Further comments

Laman-plus-two characterisation

It is straightforward to see that a similar reduction to that described in the proof of Theorem 7.2.20 may be applied to characterise all Laman-plus-two graphs, using only a set of standard inductive moves.

Theorem 7.2.25. Gis a Laman-plus-two graph if and only if G may be constructed

from a base graph B ∈ {B5, B6, B7, B8,1, B8,2, B8,3} by a sequence of 0-extensions, 1-extensions, and X-replacements applied outside the circuits of G.

7.2 Two colour classes in 2 dimensions 179

Proof. The base graphs are illustrated in Figure 7.6, and may be straightforwardly

checked to confirm that each is a Laman-plus-two graph. As 0-extensions and 1- extensions result in a larger Laman graph when applied to a Laman graph, it is clear that applying these moves to a Laman-plus-two graph will produce a Laman-plus- two graph. We note that applying X-replacements to edges outside the circuits of a Laman-plus-two graph also results in a larger Laman-plus-two graph. Therefore any graph constructed from a base graph B ∈ {B5, B6, B7, B8,1, B8,2, B8,3} by applying 0-extensions and 1-extensions, and by applying X-replacements to pairs of disjoint edges that lie outside the circuits of the smaller graph, will be a Laman-plus-two graph.

To prove that any Laman-plus-two graph may be constructed in this way, we apply induction on the number of vertices. This proof follows the same structure as the proof of Theorem 7.2.20. Let G= (V, E) be a Laman-plus-two graph that is not one of the

base graphs.

The Laman-plus-two graphG will contain at least two circuits. Let two of these

circuits be labelled C1 and C2. Removing any edge e1 ∈E(C1)\E(C2) results in a Laman-plus-one graph that still contains the circuit C2, so C2 is the unique circuit within G−e1. Any edge e2 ∈ E(C2)\E(C1) may be removed to leave the Laman graph G∗ =G− {e1, e2}.

If G contains any vertices of degree 2, they will lie outside all circuits within G.

The vertex and its pair of edges may be removed without changing the circuits ofG, so

the reduced graph G′ will remain as a Laman-plus-two graph. By a similar argument

to that seen in the proof of Theorem 7.2.20, any vertex of degree 3 that lies outside the circuits of the Laman-plus-two graph G may be reduced while leaving the circuits

unchanged, and so this 1-reduction will result in a smaller Laman-plus-two graph. By Proposition 7.2.3, if the Laman-plus-two graphG contains two disjoint circuits

7.2 Two colour classes in 2 dimensions 180 the circuits of G will have degree exactly 4. The section “Vertices of degree 4” of the

proof of Theorem 7.2.20 (page 160) shows that in such a situation, there will be a vertex of degree 4 outside the circuits of G that may be removed and replaced by a

pair of edges with distinct end points, such that the reduced graph G′ will also be a

Laman-plus-two graph.

It remains to show that when every vertex of the Laman-plus-two graph G is

within a circuit, the reverse of an inductive move may be applied to create a smaller Laman-plus-two graph. As the minimum degree within a circuit is 3, the minimum degree within G is 3.

There will either be exactly two edge-disjoint circuits within G, at least one of

which has |V(C)| ≥5, or there will be two circuits with E(C1)∩E(C2)̸=∅. All other circuits within Gwill be contained within C1 ∪C2, by Proposition 7.2.2a, and so we may suppose that any other circuit C3 has |V(C3)| ≥ |V(C1)|, |V(C3)| ≥ |V(C2)|, and that at least one of C1 and C2 has |V(C)| ≥ 5. Without loss of generality, suppose that |V(C1)| ≥5.

The Laman-plus-two graph G contains at least two vertices with deg_G(x) = 3,

and a circuit contains at least four vertices with degC1(x) = 3. If Gcontains exactly

two edge-disjoint circuits, there will be either two or three vertices within C1 with degG(v)>degC1(v), and so there will be at least one vertex with degG(x) = degC1(x) =

3. If instead C1 has a non-trivial intersection with C2, and C2 is a copy ofK4, there will also be either two or three vertices with degG(v) > degC1(v), and at least one

vertex in V(C1)\V(C2) of degree 3. From Claim 7.2.21, if neither of C1 and C2 is a copy of K4 and |V(C1 ∩C2)| ≥ 2, there is at least one vertex of degree 3 within

V(C1)\V(C2).

Removing the vertex x ∈ V(C1) with degG(x) = degC1(x) = 3, and its three

7.2 Two colour classes in 2 dimensions 181 which is a Laman-plus-one graph. Since the removed vertex was not within a copy of K4, at least one pair of neighbours of this vertex will not be joined by an edge in

b

G. We may add this edge to complete the reverse of the 1-extension, and create a

Laman-plus-two graph G′.

Example 7.2.26. Figure 7.29 shows a Laman-plus-two graph Gwhere every vertex

lies within at least one circuit. Three circuits C1, C2, C3 are illustrated below, and it is straightforward to see that any circuit lies within the union of the other two (Proposition 7.2.2a). Each of the vertices labelled v1, v2 andv3 has degree 3 and the reverse of a 1-extension may be applied at any of these vertices. We note that removing the vertex vi and its three associated edges from Gresults in precisely the circuit Ci

(1≤i≤3).

v2

v1

v3

a The Laman-plus-two graph, G. b The circuit C1.

c The circuitC2. d The circuit C3.

7.3 Higher k in 2 dimensions 182