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Deterministic Soliton Graphs

Miklós Bartha

Department of Computer Science Memorial University of Newfoundland St. John’s, NL, Canada

E-mail: bartha@cs.mun.ca

Miklós Krész

Department of Computer Science Juhász Gyula Teacher Training College University of Szeged, Hungary

E-mail: kresz@jgytf.u-szeged.hu

Keywords:soliton automata, matching theory

Received:March 23, 2005

Soliton graphs are studied in the context of a reduction procedure that simplifies the structure of graphs without affecting the deterministic property of the corresponding automata. It is shown that an elementary soliton graph defines a deterministic automaton iff it reduces to a graph not containing even-length cycles. Based on this result, a general characterization is given for deterministic soliton graphs using chestnut graphs, generalized trees, and graphs having a unique perfect matching.

Povzetek: ˇClanek obravnava grafe brez lihih ciklov.

1 Introduction

One of the most ambitious goals of research1 in modern bioelectronics is to develop a molecular computer. The in-troduction of the concept “soliton automaton” in [5] has been inspired by this research, with the intention to cap-ture the phenomenon called soliton waves [4] through an appropriate graph model.

Soliton graphs and automata have been systematically studied by the authors on the grounds of matching theory in a number of papers. Perhaps the most significant contri-bution among these is [2], where soliton graphs have been decomposed into elementary components, and these com-ponents have been grouped into pairwise disjoint families based on how they can be reached by alternating paths start-ing from external vertices. This paper can also serve as a source of further references on soliton automata for the in-terested reader.

Since soliton automata are proposed as switching de-vices, deterministic automata are in the center of investi-gations. The results reported in this paper are aimed at pro-viding a complete characterization of deterministic soliton automata. The two major aspects of this characterization are:

1. Describing elementary deterministic soliton graphs. 2. Recognizing that deterministic soliton graphs hav-ing an alternathav-ing cycle follow a simple hierarchical pattern called a chestnut.

1Partially supported by Natural Science and Engineering Research

Council of Canada, Discovery Grant #170493-03

An important tool in the study of both aspects is a reduction procedure, which might be of interest by itself. It allows elementary deterministic soliton graphs to be reduced to generalized trees, and it can also be used to reduce chestnut graphs to really straightforward ones, called baby chestnuts.

2 Soliton graphs and automata

By a graph G = (V(G), E(G)) we mean an undirected graph with multiple edges and loops allowed. A vertexv∈ V(G)is calledexternalif its degree is one, andinternalif the degree ofvis at least two. An internal vertex isbaseif it is adjacent to an external one. External edges are those of

E(G)that are incident with at least one external vertex, and internal edges are those connecting two internal vertices. GraphGis calledopenif it has at least one external vertex. A walk in a graph is an alternating sequence of ver-tices and edges, which starts and ends with a vertex, and in which each edge is incident with the vertex immediately preceding it and the vertex immediately following it. The

lengthof a walk is the number of occurrences of edges in

it. A trailis a walk in which all edges are distinct and a

pathis a walk in which all vertices are distinct. A cycle is a trail which can be decomposed into a path and an edge connecting the endpoints of the path. Ifα=ve1. . . enwis a walk fromv towandβ = wf1. . . fkzis a walk from w to z, then the concatenation of α and β is the walk

αkβ=ve1. . . enwf1. . . fkzfromvtoz.

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no vertex of Goccurs more than once as an endpoint of some edge in M. It is understood by this definition that loops cannot participate in any matching. The endpoints of the edges contained in M are said to be covered by

M. A perfect internal matching is a matching that

cov-ers all of the internal vertices. An edgee E(G)is

al-lowed (mandatory)ifeis contained in some (respectively,

all) perfect internal matching(s) ofG.Forbidden edgesare those that are not allowed. We shall also use the term

con-stant edge to identify an edge that is either forbidden or

mandatory. An open graph having a perfect internal match-ing is called asoliton graph. A soliton graphGis elemen-taryif its allowed edges form a connected subgraph cover-ing all the external vertices. Observe that ifGis elemen-tary, then it cannot contain a mandatory edge, unlessGis a mandatory edge by itself with a number of loops incident with one of its endpoints.

LetGbe an elementary soliton graph, and define the re-lationonInt(G)as follows: v1 v2if an extra edge econnectingv1withv2becomes forbidden inG+e. It is known, cf. [6, 2], thatis an equivalence relation, which determines the so calledcanonical partitionof (the internal vertices of)G. The reader is referred to [6] for more infor-mation on canonical equivalence, and on matching theory in general.

LetGbe a graph andM be a matching ofG. An edge

e∈E(G)is said to beM-positive(M-negative) ife∈M

(respectively, e 6∈ M). AnM-alternating path (cycle)in

G is a path (respectively, even-length cycle) stepping on

M-positive andM-negative edges in an alternating fash-ion. AnM-alternating loopis an odd-length cycle having the same alternating pattern of edges, except that exactly one vertex has two negative edges incident with it. Let us agree that, if the matchingM is understood or irrelevant in a particular context, then it will not be explicitly indicated in these terms. Anexternal alternating pathis one that has an external endpoint. If both endpoints of the path are ex-ternal, then it is called acrossing. An alternating path is

positiveif it is such at its internal endpoints, meaning that

the edges incident with those endpoints are positive. LetGbe a soliton graph, fixed for the rest of this sec-tion, and let M be a perfect internal matching ofG. An

M-alternating unitis either a crossing or an alternating

cy-cle with respect to M. Switchingon an alternating unit amounts to changing the sign of each edge along the unit. It is easy to see that the operation of switching on anM -alternating unitαcreates a new perfect internal matching

S(M, α)for G. Moreover, as it was proved in [1], every perfect internal matchingM ofGcan be transformed into any other perfect internal matching M0 by switching on a collection of pairwise disjoint alternating units. Conse-quently, an edgeeofGis constant iff there is no alternat-ing unit passalternat-ing throughewith respect to any perfect in-ternal matching ofG. A collection of pairwise disjointM -alternating units will be called anM-alternating network, and the network transforming one perfect internal matching

Minto anotherM0will be denoted byN(M, M0). Clearly,

a) c-trail b) l-trail

Figure 1: Soliton trails.

N(M, M0)is unique.

Now we generalize the alternating property to trails and walks. Analternating trailis a trailαstepping on positive and negative edges in such a way thatαis either a path, or it returns to itself only in the last step, traversing a negative edge. The trail αis a c-trail(l-trail) if it does return to itself, closing up an even-length (respectively, odd-length) cycle. That is,α=α1 2, whereα1is a path andα2is a cycle. These two components ofαare called thehandle

andcircuit, in notation,h(α)andc(α). The joint vertex

on h(α)andc(α) is called thecenter of α. Anexternal

alternating trailis one starting out from an external vertex,

and asoliton trailis a proper external alternating trail, that is, either a c-trail or an l-trail. See Fig. 1.

The collection of external alternating walksinGwith respect to some perfect internal matchingM, and the con-cept of switching on such walks are defined recursively as follows.

(i) The walkα=v0ev1, wheree= (v0, v1)withv0being external, is an externalM-alternating walk, and switching onαresults in the setS(M, α) =M∆{e}. (The operation

∆is symmetric difference of sets.)

(ii) Ifα=v0e1. . . envnis an externalM-alternating walk ending at an internal vertex vn, and en+1 = (vn, vn+1) is such that en+1 S(M, α) iff en S(M, α), then α0 = αe

n+1vn+1 is an external M-alternating walk and

S(M, α0) =S(M, α)∆{e n+1}.

It is required, however, that en+1 6= en, unless en S(M, α)is a loop.

It is clear by the above definition thatS(M, α)is a perfect internal matching iff the endpointvnofαis external, too. In this case we say thatαis asoliton walk.

EXAMPLE Consider the graph G of Fig. 2, and let

M = {e, h1, h2}. A possible soliton walk from utov with respect toM isα=uewgz1h1z2l2z3h2z4l1z1gwf v. Switching onαthen results inS(M, α) ={f, l1, l2}.

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h

Figure 2: Example soliton graphG

by

δ(M,(v, w)) ={S(M, α)|αis a soliton walk,vtow}.

GraphGis calleddeterministicifAG is such in the usual sense, that is, if for every stateM and input(v, w),

(M,(v, w))| ≤1.

EXAMPLE (Continued) Observe that the soliton

au-tomaton defined by the graph of Fig. 2 is non-deterministic, asα0 =uewf vis also a soliton walk fromutovwith re-spect to stateM such thatS(M, α)6=S(M, α0).

Letαbe a soliton c-trail with respect toM. It is easy to see that the walk

s(α) =h(α)kc(α)kh(α)R

is a soliton walk, and the effect of switching ons(α)is the same as switching on the cyclec(α)alone. (For any walk

β,βRdenotes the reverse ofβ.) Ifαis a soliton l-trail, then s(α)is defined as the soliton walk

s(α) =h(α)kc(α)kc(α)kh(α)R.

Clearly, this walk induces a self-transition ofAG, that is, no state change is observed. In the sequel, all perfect internal matchings of Gwill simply be called states for obvious reasons.

Recall from [5] that an edgeeofGisimperviousif there is no soliton walk passing throughein any state ofG. Edge

eisviableif it is not impervious. See Fig. 2, edgehfor an

example of an impervious edge.

We are going to give a simpler characterization of imper-vious edges in terms of alternating paths, rather than walks. To this end, we need the following lemma.

Lemma 2.1. Ifαis an externalM-alternating walk from

v tou, then there exists anM-alternating network Γand

an externalM-alternating trailβfromvtousuch that

1.Γconsists of cycles only, and it is disjoint fromβ;

2. S(M, α) =S(S(M,Γ), β).

Proof. Easy induction on the length of α, left to the reader. 2

An internal vertex v V(G) is called accessible with

respect to state M if there exists a positive external M -alternating path leading tov. It is easy to see, cf. [2], that vertexvis accessible with respect to some stateM iffvis accessible with respect to all states ofG.

Proposition 2.2. For every edgee∈E(G),eis impervious

iff both endpoints ofeare inaccessible.

Proof. If either endpoint of e is accessible, then e is clearly viable. Assume therefore that both endpoints u1 and u2 of e are inaccessible, and let α be an arbitrary external M-alternating walk fromv Ext(G)to either

u1 or u2, sayu1. By Lemma 2.1, there exists a suitable externalM-alternating trailβ fromv tou1. Each internal edge lying onβ has an accessible endpoint, so thateis not among them. Moreover, the edge ofh(β)incident withu1 must be positive with respect toS(M, α), otherwiseh(β)

would be a positive external alternating path with respect to stateS(M,Γ). (Recall thath(β)is the handle ofβ, and take h(β) = β if β is just a path.) But then emust be negative with respect toS(M, α), or elseh(β)eu2 would be a positive external alternating path leading tou2 (with respect to S(M,Γ), or even M, since Γ is disjoint from

β). We conclude that the walkα cannot continue one, because it must take the two positive edges incident with

u1before and after hitting that vertex. Thus, every time an external alternating walk reaches either endpoint of e, it will misseas a possible continuation. In other words,eis impervious. 2

3 Elementary decomposition of

soliton graphs

Again, let us fix a soliton graph Gfor the entire section. In general, the subgraph of Gdetermined by its allowed edges has several connected components, which are called

theelementary componentsofG. Notice that an

elemen-tary component can be as small as a single external ver-tex ofG. Such elementary components are called degener-ate, and they are the only exception from the general rule that each elementary component is an elementary graph.

Amandatory elementary componentis a single mandatory

edge e E(G), which might have a loop around one or both of its endpoints.

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D

Figure 3: Elementary components in a soliton graph

of course). A viable elementary component istwo-wayif it is not one-way. Animperviouselementary component is one that is not viable.

EXAMPLE The graph of Fig. 3 has five elementary

components, among which C1 andD are external, while C2, C3 andC4 are internal. Component C3 is one-way with the canonical class{u, v}being principal, whileC2is two-way andC4is impervious.

LetC be an elementary component ofG, andM be a state. AnM-alternatingC-earis a negativeM-alternating path or loop having its two endpoints, but no other vertices, inC. The endpoints of the ear will necessarily be in the same canonical class ofC. We say that elementary com-ponentC0 istwo-way accessiblefrom componentCwith respect to any (or all) state(s)M, in notationCρC0, ifC0is covered by anM-alternatingC-ear. It is required, though, that ifCis one-way and internal, then the endpoints of this

earnot bein the principal canonical class ofC. As it was

shown in [2], the two-way accessible relationship is match-ing invariant. Afamilyof elementary components inGis a block of the partition induced by the smallest equivalence relation containing ρ. A family F is called external if it contains an external elementary component, otherwise F

is internal. A degenerate familyis one that consists of a single degenerate external elementary component. Family

F isviableif every elementary component inF is such,

andFisimperviousif it is not viable. As it turns out

eas-ily, the elementary components of an impervious family are all impervious. Soliton graphGisviableif all of its fami-lies are such.

EXAMPLE(CONTINUED) Our example graph in Fig. 3 has four families: F1 = {C1, C2},F2 = {D},F3 =

{C3},F4={C4}. FamilyF1is external,F2is degenerate, andF3 is internal. These families are all viable, whereas familyF4is impervious.

The first group of results obtained in [2] on the structure of elementary components ofGcan now be stated as follows.

Theorem 3.1. Each viable family ofGcontains a unique

one-way elementary component, called the root of the fam-ily. Each vertex in every member of the family, except for the principal vertices of the root, is accessible. The princi-pal vertices themselves are inaccessible, but all other ver-tices are only accessible through them.

A familyFis callednear-externalif each forbidden vi-able edge incident with any principal vertex of its root is external. For two distinct viable familiesF1andF2,F2is said to followF1, in notationF1 7→ F2, if there exists an edge inGconnecting any non-principal vertex inF1 with a principal vertex of the root ofF2. The reflexive and tran-sitive closure of7→is denoted by7→∗. The second group of results in [2] characterizes the edge connections between members inside one viable family, and those between two different families.

Theorem 3.2. The following four statements hold for the

families ofG.

1. An edge einside a viable family F is impervious

iff both endpoints ofeare in the principal canonical

class of the root. Every forbidden edgeeconnecting

two different elementary components in F is part of

an ear to some memberC∈ F.

2. For every edgeeconnecting a viable familyF1to

any other family (viable or not)F2, at least one

end-point ofeis principal inF1orF2. If the endpoint ofe

inF1is not principal, thenF2is viable and it follows

F1.

3 The relation7→∗ is a partial order between viable

fam-ilies, by which the external families are minimal ele-ments.

4 If F and G are families such that F 7→ G, then∗

each non-principal vertexuofGis accessible fromF,

meaning that for every stateM there exists a positive

M-alternating path toueither from a suitable

exter-nal vertex ofF, ifFis external, or from an arbitrary

principal vertex ofF, ifFis internal. The pathαruns

entirely in the families betweenFandGaccording to

7→.

For convenience, the inverse of the partial order7→∗ will be referred to as ≤G. Theorems 3.1 and 3.2 are funda-mental regarding the structural decomposition of soliton graphs, and they will be applied liberally throughout the forthcoming sections (especially in Section 4). There will be no explicit reference made, however, to these theorems whenever they apply.

4 Chestnuts

Chestnuts have been introduced in [5] as a group of de-terministic soliton graphs having a very simple and easily recognizable structure.

Definition 1.A connected graphGis called achestnutif it

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Figure 4: A chestnut.

(i) V(αi)∩V(αj) =for1≤i6=j≤k;

(ii) V(αi)∩V(γ)consists of a unique vertex – denoted byvi– for eachi∈[k];

(iii) vi andvj are at even distance on γ for any distinct i, j∈[k];

(iv) any vertex wi V(αi)withd(wi) > 2 is at even distance fromviinαifor eachi∈[k].

See Fig. 4 for an example chestnut.

Our first observation regarding chestnuts is that they are bipartite graphs. Let us call a vertex of a chestnutGouter

if its distance from any of thevi’s is even, andinnerif this distance is odd. Then the inner and outer vertices indeed define a bipartition of G. Moreover, the degree of each inner vertex is at most 2. It is easy to come up with a per-fect internal matching forG: just mark the cycle γin an alternating fashion, then the continuation is uniquely deter-mined by the structure of the treesαi. Thus,Ghas exactly two states. It is also easy to see that the inner internal ver-tices are accessible, while the outer ones are inaccessible. Thus, the cycleγforms an internal elementary component with its outer vertices constituting the principal canonical class of this component. Moreover,γforms a stand-alone internal family inG. The rest ofG’s families are all single mandatory edges along the trees αi, or they are degener-ate ones consisting of a single inner external vertex. Their rank in the partial order≤G is consistent with their posi-tion in the respective treesαi, following a decreasing order from the leafs to the root. The family{γ}is the minimum element of≤G, andGhas no impervious edges.

By the description above, every chestnut Gis a deter-ministic soliton graph. Moreover,Gis strongly determin-istic [5] in the sense that, for each pair(v1, v2)of external vertices, there exists at most one soliton walk fromv1to v2 in each state ofG. We are going to show that for ev-ery connected soliton graphGhaving no impervious edges, but possessing a non-mandatory internal elementary com-ponent,Gis deterministic iffGis a chestnut.

Lemma 4.1. Letαbe an external M-alternating path of

a soliton graphGleading to a principal vertexvin some

internal family F. Then αcan be extended to a soliton

trail, the center of which lies in a familyH ≤GF.

Proof. Every possible continuation β of α as an M -alternating path can only leave the family F by entering another family G <G F. It is therefore inevitable that, whenβfinally returns to itself, this will happen in a family

H ≤GF. The pathβmust eventually return to itself, since Gis finite. 2

Lemma 4.2. Letβ be a soliton c-trail of a deterministic

soliton graphGstarting fromv∈Ext(G)with respect to

stateM. Then, starting fromv, there exists no soliton trail

with respect toM that is different fromβ.

Proof.Assume, by contradiction, that an unwantedδ6= βexists. Clearly,c(δ) =c(β), otherwise the soliton walks

s(β)ands(δ)would define two different transitions ofAG in stateMon input(v, v). Therefore we haveh(β)6=h(δ). Starting fromv, letzbe the first vertex on bothh(β)and

h(δ)where these paths split into two different directions (or just use a pair of parallel edges to reach the same vertex). Thus,β =γkβ0andδ=γkδ0, whereγis a suitable path fromv toz, andβ0,δ0 are M-alternating c-trails starting out fromzon different edges. Clearly, the last edge of γ

incident withzis positive, and the first edges ofβ0 andδ0 incident withzare negative. Therefore the walk

χ=γkβ0kh(β0)Rkδ0kh(δ0)RkγR

is a soliton walk fromvtov, which defines a self-transition ofAG. This transition, however, is different from the one defined by the walkss(β)ands(γ); a contradiction. 2

Theorem 4.3. Let G be a deterministic soliton graph

with no impervious edges, and assume thatGhas a

non-mandatory elementary component C lying in an internal

family F. Then Cconsists of a single even-length cycle,

F = {C}andF is a minimal element with respect to to

the partial order≤G.

Proof.LetM be an arbitrary state ofG, andαbe a neg-ative externalM-alternating path from somev ∈Ext(G)

to a principal vertexzof the rootRofF. Furthermore, let

γ be anM-alternating cycle inC. SinceCis accessible throughz, we can fix a soliton c-trailβwith respect toM

such thatβ starts out fromv andc(β) = γ. We can as-sume, without loss of generality, thatR=C. For, we need to rule out the only possible scenario that is incompatible with this assumption, namely whenRis a single mandatory edgee= (z, w). Since in this case there are two-way mem-bers of the familyF(e.g.C), there exists anM-alternating

R-ear (loop)²aroundw. The loop²gives rise to a soliton l-trailδ=αe², which, by Lemma 4.2, cannot co-exist with

β.

Now let us assume that R = C has an allowed edge different from the ones alongγ. Clearly,Cthen has another

M-alternating cycle γ0 6= γ. As above, it is possible to extendαto a soliton c-trailβ0with respect toM such that

c(β0) = γ0. Again, this contradicts Lemma 4.2, knowing thatβ 6=β0.

We have seen so far thatR = C, andC is spanned by

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Also notice that vertexzonγlies outside one ofχ1andχ2. Consequently,αcan again be extended to a soliton l-trailδ

such thatc(δ) =χ1orc(δ) =χ2. (Remember thatzand u1(2) are in different canonical classes.) A contradiction with Lemma 4.2 is reached, showing thatC=γ.

It remains to be seen thatF = {C}, andF is minimal with respect to≤G. AnyM-alternatingC-ear² originat-ing from two different non-principal verticesu1andu2is equivalent to an edge connectingu1directly withu2, and so need not be considered separately. On the other hand, if the ear²was anM-alternating loop, thenαcould again be trivially extended to a soliton l-trail δ withc(δ) = ², which is impossible. Now assume that there exists a fam-ilyG <G F, and continueαto obtain a negative external M-alternating path α0 leading to a principal vertex of the root ofG. By Lemma 4.1,α0 can further be extended to a soliton trailδhaving its center in a familyH ≤ G. The trail

δis therefore different from β, which contradicts Lemma 4.2. The proof is now complete. 2

Theorem 4.4. Let G be a connected deterministic

soli-ton graph having no impervious edges. If Ghas a

non-mandatory internal elementary component, then G is a

chestnut.

Proof.Induction on the number of non-degenerate ele-mentary components ofG. Assume that a non-mandatory internal elementary componentCexists inG. IfCis the only non-degenerate elementary component inG, thenC

is an internal family by itself, and the statement of the the-orem follows directly from Thethe-orem 4.3. Now letGhave more than one non-degenerate elementary component, and suppose that the statement is true for all appropriate soli-ton graphs having fewer non-degenerate elementary com-ponents thanG. LetFdenote the family ofC. By Theo-rem 4.3, ifF is internal, thenF ={C}andFis minimal with respect to≤G. LetGbe a non-degenerate family such that F ≤G G, and G is either external or near-external. Clearly, ifF is external, thenG = F. OtherwiseG can be found by stepping upwards in the partial order≤G start-ing from familyF, which is a minimal element by The-orem 4.3. Notice that this search must rejectF itself, as

F being near-external would imply that its sole member

Cis the only non-degenerate elementary component inG. Thus, in this case,G 6=F.

LetRbe the root ofG. We are going to prove that 1. Ris mandatory,

2. G={R}, and

3. there is exactly one forbidden edge incident withR’s unique non-principal (or non-external) vertex.

Fix a stateM forG, and choose anM-alternating cycle

γinC. SinceCis accessible fromG,γcan be extended to a soliton c-trailβ fromv Ext(G), wherevis either in

R, or it is adjacent to a principal vertex inR.

Proof of Statement 1. Assume, by contradiction, thatR

is non-mandatory, and distinguish the following two cases.

Case 1: Ris external. Letube the vertex inR where

the pathh(β)finally leaves this elementary component. By

[1], there exists a crossingδinRfromvto another external vertexzviauwith respect to some stateM¯RofR. Modify M, so that its restriction toRis replaced byM¯R, and let M0denote the resulting state. Clearly, the straight crossing

δbetweenvandzin either direction is a possible transition ofAGin stateM0. On the other hand, this crossing can also make a detour to includeγthrough the appropriate section of β that starts atu. Notice, however, that this detour is only available from one direction, depending on whether theM0-positive edge incident withuonδpoints towardv orz. Nevertheless, the co-existence of these two different transitions violates the deterministic property.

Case 2: Ris internal. Trivially, there exists a soliton

c-trailδwith respect toM starting fromv such thatc(δ)

runs entirely inR; a contradiction with Lemma 4.2.

Proof of Statement 2. As in the proof of Theorem 4.3,

the existence of anR-ear as anM-alternating loop would immediately contradict Lemma 4.2.

Proof of Statement 3. Assume that there is more than

one forbidden edge going out from the non-principal (or non-external) vertex of G to different internal families of

G. By Lemma 4.1, each of these edges can be made part of a suitable soliton trail inGwith respect toM, starting from vertexv. Sinceβ is also such a trail, a contradiction with Lemma 4.2 is inevitable.

Now we are ready to synthesize statements 1, 2, 3, and finish the proof of Theorem 4.4. It has turned out that the caseF =Gis not possible. Detach the mandatory family

GfromG, keeping the unique forbidden edge specified in statement 3 as an external edge in the remainder graphG0. Notice that, if R is internal, then its principal vertex can only be adjacent to external vertices, or elseGwould have impervious edges. Observe that G0 is also deterministic, connected, has no impervious edges, and still has the non-mandatory internal elementary component C in it. Apply the induction hypothesis to establish G0 as a chestnut. Finally, conclude that Gis also a chestnut by sticking back the mandatory familyGontoG0. The proof is complete. 2

5 Reducing soliton graphs

Aredexrin graphGconsists of two adjacent edgese =

(u, z)andf = (z, v)such that u 6= v are both internal and the degree ofzis 2. The vertexzis called thecenter

of r, whileuand v (e andf) are the twofocal vertices (respectively,focaledges) ofr.

Letrbe a redex in G. ContractingrinGmeans cre-ating a new graphGrfromGby deleting the center ofr and merging the two focal vertices ofrinto one vertexs. Now suppose thatGis a soliton graph. For a stateM ofG, letMrdenote the restriction ofMto edges inGr. Clearly, Mr is a state ofGr. Notice that the stateM can be re-constructed fromMrin a unique way. In other words, the connectionM 7→ Mr is a one-to-one correspondence be-tween the states ofGand those ofGr.

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of αto edges inGr. It is easy to see that ifαis a soli-ton walk in G with respect to M, then so is tracer(α) inGr with respect toMr. Moreover, the soliton walkα can again be uniquely recovered fromtracer(α). Conse-quently, the connectionα 7→ tracer(α)is also a one-to-one correspondence between soliton walks inGand soli-ton walks inGr. Furthermore,M0 =S(M, α)holds inG iff(M0)

r =S(Mr,tracer(α))holds inGr. We thus have

proved the following statement.

Proposition 5.1. The soliton automataAG and AGr are

isomorphic.

Notice, furthermore, that if an alternating unit goes through both focal vertices ofr, then it must do so along the center ofr. As a consequence we have:

Proposition 5.2. The function tracerestablishes a

one-to-one correspondence between the alternating units ofGand

those ofGr.

It follows from the previous two propositions that every edgeeofGris allowed inGriffeis allowed inG. As to the two focal edges ofr, they can either be allowed or not inG, even whenGris elementary. This issue is addressed by Proposition 5.3 below.

Proposition 5.3. Letrbe a redex in soliton graphG, and

assume that Gr is elementary. Then Gis elementary iff

both focal edges ofr are allowed in G, or, equivalently,

each focal vertex ofrhas at least one allowed edge ofGr

incident with it.

Proof. It is sufficient to note that either focal edge of

ris forbidden inGiff the other focal edge is mandatory. Moreover, an arbitrary internal edgeeof Gis mandatory iff all edges adjacent toeare forbidden. 2

Another natural simplifying operation on graphs is the removal of a loop from around a vertexv if, after the re-moval, v still remains internal. Such loops will be called

secondary. LetGv denote the graph obtained fromGby

removing a secondary loopeat vertexv. Clearly, ifGis a soliton graph, then so isGv, and the states ofGvare ex-actly the same as those ofG. The automataAGandAGv,

however, need not be isomorphic. This follows from the fact that any external alternating walk reachingvon a pos-itive edge can turn back inGafter having made the loop

etwice, while this may not be possible for the same walk without the presence ofe. Nevertheless, it is still true that for every elementary soliton graphG,Gis deterministic iff

Gvis such.

There are loops, however, the removal of which pre-serves isomorphism of soliton automata. These loops are exactly the ones around the inaccessible vertices of G. Each such loop is impervious, so that its removal does not affect the automaton behavior ofG.

6 General characterization of

deterministic soliton graphs

GraphGis said to bereducedif it is free from redexes and secondary loops. Ageneralized treeis a connected graph not containing any even-length cycles. By this definition, the odd-length cycles possibly present in a generalized tree must be pairwise edge-disjoint, which explains the termi-nology.

The proof of the following statement is left to the reader as a simple exercise.

Lemma 6.1. Letαbe an alternating cycle with respect to

stateM of an elementary soliton graphG. ThenGhas an

M-alternating cycleα0and a crossingβthat intersectsα0.

Proposition 6.2. IfGis nondeterministic, thenGhas an

alternating cycle with respect to some stateM. Conversely,

ifGis elementary and has an alternating cycle with respect

to some stateM, thenGis nondeterministic.

Proof.Assume thatGis nondeterministic. Then there exists a stateMand soliton walksα,βconnecting the same pair of external vertices in such a way that S(M, α) 6= S(M, β). Consider the network N(S(M, α), S(M, β)). This network is not empty, and it consists of alternating cycles only.

Now letGbe elementary, having anM-alternating cycle

α. By Lemma 6.1 we can assume thatGalso has a crossing

β with respect to the same stateM that intersectsα. Con-sider the network N(S(M, β), S(M, α)). This network will contain a crossingβ0different fromβ, yet connecting the same two external verticesv1, v2. Thus, for the state M0 =S(M, β),A

Ghas two different transitions on input

(v1, v2)resulting in statesS(M0, β0)andS(M0, β) =M, respectively. 2

The key step to our results in this section is Theorem 6.3 be-low. The proof of this theorem is rather complex, therefore we do not present it here. The interested reader is referred to [3] for a complete proof.

Theorem 6.3. Let G be a reduced elementary soliton

graph. IfGcontains an even-length cycle, then it also has

an alternating cycle with respect to some state ofG.

For an arbitrary graph G, contract all redexes and re-move all secondary loops in an iterative manner to obtain a reduced graphr(G). Observe that this reduction proce-dure has the so called Church-Rosser property, that is, if

Gadmits two different one-step reductions to graphs G1 andG2, then eitherG1is isomorphic toG2, orG1andG2 can further be reduced to a common graph G1,2. In this context, one reduction step means contracting a redex or removing a single secondary loop. As an immediate con-sequence of the Church-Rosser property, the graph r(G)

(8)

the graph obtained fromGafter i-reduction is denoted by

ri(G).

Theorem 6.4. For any graph G, ifr(G)is a generalized

tree, thenGis a deterministic soliton graph. Conversely, if

Gis a deterministic elementary soliton graph, thenr(G)is

a generalized tree.

Proof. Clearly, G is a soliton graph iffr(G) is such. By Proposition 5.2, if r(G)is a generalized tree, thenG

does not contain alternating cycles with respect to any of its states. Proposition 6.2 then implies thatGis determin-istic. Conversely, if Gis a deterministic elementary soli-ton graph, then so isr(G), containing no alternating cycles with respect to any of its states. (See again Propositions 5.2 and 6.2.) Thus, by Theorem 6.3,r(G)is a generalized tree.

2

Corollary 6.5. An elementary soliton graph is determinis-tic iff it reduces to a generalized tree.

Definition 2.Ababy chestnutis a chestnutγ+α1+. . .+

αk such thatγ is a pair of parallel edges and each treeαi (1≤i≤k) consists of one edge or two adjacent edges.

Theorem 6.6. letGbe a viable connected soliton graph possessing a non-mandatory internal elementary

compo-nent. ThenGis deterministic iffri(G)is a baby chestnut.

Proof.‘Only if’ By Theorem 4.4,Gis a chestnut aug-mented by some impervious edges connecting the outer in-ternal vertices with each other. Since each inin-ternal inner vertex, different from the base ones, is the center of a re-dex, we can eliminate all of these vertices using reduction, except of course the last inner vertex in γ, which will no longer identify a redex. After removing the secondary im-pervious loops generated during redex elimination, ri(G) becomes a baby chestnut.

‘If’ Blowing up γ by inverse redex elimination, or stretching the treesαiin this manner preserves the property of being a chestnut, and any impervious loops added dur-ing this procedure may only stretch into impervious edges. Thus, the graph resulting from a baby chestnut after any number of blow-ups and stretches is still a chestnut with some additional impervious edges. 2

Now we are ready to state the main result of this paper.

Theorem 6.7. LetGbe a connected viable soliton graph.

ThenGis deterministic iff it satisfies one of the following

two conditions.

1. Gi-reduces to a baby chestnut.

2. Each external component of Greduces to a

general-ized tree, and the subgraph ofGdetermined by its internal

components has a unique perfect matching.

Proof. Immediate by Theorems 4.4, 6.6, and Corol-lary 6.5. 2

7 Conclusion

We have presented a detailed analysis of deterministic soliton graphs. First we proved that every connected deterministic soliton graph having no impervious edges, but possessing a non-mandatory internal elementary component is a chestnut. Then we introduced a simple reduction procedure on graphs, and showed that an elementary soliton graph is deterministic iff it reduces to a generalized tree. Using i-reduction, we could provide a yet simpler description of chestnut graphs, and gave a characterization of deterministic soliton graphs in general.

References

[1] M. Bartha, E. Gombás, On graphs with perfect internal matchings,Acta Cybernetica12(1995), 111–124.

[2] M. Bartha, M. Krész, Structuring the elementary com-ponents of graphs having a perfect internal matching,

Theoretical Computer Science299(2003), 179–210.

[3] M. Bartha, M. Krész, Deterministic soliton automata defined by elementary graphs, inKalmár Workshop on

Logic and Computer Science, Technical Report,

Uni-versity of Szeged, 2003, pp. 69–79.

[4] F. L. Carter, A. Schultz, D. Duckworth, Soliton switch-ing and its implications for molecular electronics, in

Molecular Electronic Devices II (F. L. Carter Ed.),

Marcel Dekker Inc., New york, 1987, pp. 149–182.

[5] J. Dassow, H. Jürgensen, Soliton automata,J. Comput.

System Sci.40(1990), 154–181.

Figure

Figure 3: Elementary components in a soliton graph
Figure 4: A chestnut.

References

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