(c) Maze+ feedback.
Figure 3.1 . Basic mappings.
3. **Edge**-**disjoint** Hamiltonian cycles in a 2D torus. When **edge**-**disjoint** Hamiltonian cycles are used in a communication algorithm, their eﬀectiveness is improved if more than one cycle exists. As mentioned earlier, the existence of **disjoint** Hamiltonian cycles in the cross-product of various **graphs** has been discussed in the literature [1, 4, 8, 9, 10, 11, 15]; however, all these methods do not give a straightforward way of generating such **disjoint** cycles. This section contains the functions that generate these **disjoint** cycles for the 2D torus.

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Abstract
The problem of finding the maximum number of vertex-**disjoint** uni-color paths in an **edge**-colored graph (called MaxCDP) has been recently introduced in literature, motivated by applications in social network analysis. In this paper we investigate how the complexity of the problem depends on graph parameters (namely the number of vertices to remove to make the graph a collection of **disjoint** paths and the size of the vertex cover of the graph), which makes sense since **graphs** in social networks are not random and have structure. The problem was known to be hard to approximate in polynomial time and not fixed-parameter tractable (FPT) for the natural parameter. Here, we show that it is still hard to approximate, even in FPT-time. Finally, we introduce a new variant of the problem, called MaxCDDP, whose goal is to find the maximum number of vertex-**disjoint** and color-**disjoint** uni-color paths. We extend some of the results of MaxCDP to this new variant, and we prove that unlike MaxCDP, MaxCDDP is already hard on **graphs** at distance **two** from **disjoint** paths.

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Third, another shortest path is constructed be- tween the **two** endpoints in the newly obtained graph. Fourth, paths common between the **two** constructed **disjoint** paths and the cycles that do not contain both the endpoints are removed prior to constructing the **two** **disjoint** paths. On the other hand, the proposed algorithm is not Maximum-Flow computation based and the ba- sis of the algorithm is given in the form of a single lemma. It requires only the identification of link paths prior to the construction of the dis- joint paths. Therefore, the proposed algorithm is simpler and more understandable than those based on Maximum-Flow computation. In ad- dition, most solutions available in the literature [23], [24], [19], [13] suffer from being overly complex or being unfit for use in distributed applications. These drawbacks primarily stem from the adaptation of solutions to other fun- damental problems such as **edge**-**disjoint** paths, fundamental cycles (and kernel) and network flow to the solution of the node-**disjoint** paths problem. In addition, many of these solutions require the discovery of some global properties of the entire graph instead of local properties.

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while guaranteeing some properties about the paths found. The algorithm by Itai and Rodeh [21] can also be used to solve the **disjoint** paths problem with the same time complexity as ours. On the other hand, our algorithm has a very sim- ple basis given as a simple lemma and adopts an entirely different approach. The node **disjoint** paths algorithms based on Maximum-Flow computation such as Ford-Fulkerson and Suur- balle-Tarjan [19], [13] involve a number of phases after the discovery of a shortest path be- tween **two** endpoints. First, the initial graph is transformed into a new graph where arc weights and directions are recomputed. Second, each node on the shortest path is split and addition- al arcs are introduced leading to a new graph. Third, another shortest path is constructed be- tween the **two** endpoints in the newly obtained graph. Fourth, paths common between the **two** constructed **disjoint** paths and the cycles that do not contain both the endpoints are removed prior to constructing the **two** **disjoint** paths. On the other hand, the proposed algorithm is not Maximum-Flow computation based and the ba- sis of the algorithm is given in the form of a single lemma. It requires only the identification of link paths prior to the construction of the dis- joint paths. Therefore, the proposed algorithm is simpler and more understandable than those based on Maximum-Flow computation. In ad- dition, most solutions available in the literature [23], [24], [19], [13] suffer from being overly complex or being unfit for use in distributed applications. These drawbacks primarily stem from the adaptation of solutions to other fun- damental problems such as **edge**-**disjoint** paths, fundamental cycles (and kernel) and network flow to the solution of the node-**disjoint** paths problem. In addition, many of these solutions require the discovery of some global properties of the entire graph instead of local properties. These impose severe restrictions to the adapta- tion of the solutions to distributed applications. Therefore, it is not clear how these solutions could be used to devise a distributed solution to the node-to-node **disjoint** paths problem.

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The main result of this section is given in Theorem 4:
Theorem 4. Any k-outerplanar graph such that the degree of each vertex is bounded by d ≥ 2 inside each block is (d 2 d e + ( k − 1 )b d 2 c ) -**edge**-outerplanar. Moreover, any k-**edge**-outerplanar graph is k-outerplanar.
Proof. The second part of Theorem 4 is obvious. We prove the first part by induction. Let G be a graph in OPBID k , d , k ≥ 2. For the proof, we can consider each block of G independently. Let B be a block of G with | B | ≥ 2, i.e., an inclusionwise maximal 2-vertex-connected component of G containing at least **two** vertices. Each vertex of B lying on the outer face of G is adjacent to exactly **two** edges of B lying on the outer face. For each such vertex, we remove the corresponding **two** edges. We repeat this until each vertex of B lying on the outer face of G has at most one neighbor among the vertices lying in B. At each iteration, for each vertex v lying on the outer face and still having at least **two** neighbors among the vertices in B, we remove **two** edges adjacent to v , so we have to do it at most d 2 times if d is even. If d is odd, then we stop when the residual deg B (v) is at most one, so we have to do it at most d − 2 1 times, i.e., at most b d

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A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. In 1967 Rosa[10] introduced the labeling method called β-valuation as a tool for decomposing the complete graph into isomorphic sub-**graphs**. Later on, this β-valuation was renamed as graceful labeling by Golomb [9]. A graceful labeling of a graph G with ‘q’ edges and vertex set V is an injection f :V(G) → {0,1,2,….q} with the property that the resulting **edge** labels are also distinct, where an **edge** incident with vertices u and v is assigned the label |f(u) – f(v)|. A graph which admits a graceful labeling is called a graceful graph. A variation of graceful labeling is odd-graceful labeling. This was introduced by Gnanajothi [8] in the year 1991. She defined a graph G with q edges to be odd-graceful if there is an injection f :V(G) → {0, 1, 2, . . . , 2q−1} such that, when each **edge** xy is assigned the label |f(x)−f(y)|, the resulting **edge** labels are {1, 3, 5, . . . , 2q−1}.

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Among those proposed interconnection networks, the hyper- cube is a popular interconnection network with many attrac- tive properties such as regularity, symmetry, small diameter, strong connectivity, recursive construction, partition ability, and relatively low link complexity [24]. The architecture of an interconnection network is usually modeled by a graph, where the nodes represent the processing elements and the edges represent the communication links. In this paper, we will use **graphs** and networks interchangeably.

Proof. The result is clearly true for k = 0; so suppose that k ≥ 1 and the result holds for k − 1. Fix a leaf vertex r. For each vertex v ̸= r, let e v be the last **edge** on the path from r to v , and let T v and T v ′ be the **two** component trees arising when e v is deleted from T , where v is in T v . Let v be a vertex at maximum distance from r such that T v has ≥ t leaves (and so v ̸= r). Then T v has ≤ ( d − 1 )( t − 1 ) leaves, and so T v ′ has ≥ t + ( k − 1 )( d − 1 )( t − 1 ) leaves. Thus by the induction assumption, T v ′ contains k **disjoint** subtrees each with at least t leaves, and we are done.

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In Case 3, by Claim 6, H can be considered as a graph obtained from a cubic auxiliary graph A in the following way: every **edge** of A is subdivided with one additional vertex.
Now we take an **edge**-quasi-coloring of A by applying Proposition 3. When we subdivide an **edge** e, the colors of the **two** new edges are defined to inherit the color of e. Clearly, this coloring yields a vertex coloring in G such that both color classes are dominating sets in G. Thus the coloring satisfies Property 1. A pair of adjacent edges in H yields a flat **edge** e in G if and only if their common endpoint is in T . The **edge** coloring of H , constructed above, ensures that e is monochromatic in the coloring of G and therefore, every flat **edge** is monochromatic. By our present assumption, G is cubic and diamond-free and consequently, it contains neither central nor side-edges.

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The methods used to study this problem are also of interest. There are **two** quite distinct phases leading to a complete solution. One is to construct maximal sets for various values, and this has been done using direct constructions, embeddings and amalgamations. The second phase is to show that no other sizes of maximal sets are possible. Sometimes this requires ad hoc arguments – otherwise known as “ fiddling around with **graphs** ”! Other times high powered graph theory must be involved, such as Tutte’s f -factor theorem. This diversity of approaches makes the problem especially interesting, and hopefully makes this worth reading!

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We were unable to ﬁnd a reference for this seemingly classical assertion.
Let us highlight **two** very important ideas which run through the proof of the reduction step for **edge** contractions (Theorem 6.1).
The ﬁrst of these is that we must restrict our attention to **graphs** whose constraint equations, both generic and specialised, have ﬁnitely many complex solutions. This form of rigidity for complex variables is zero dimensionality, and its signiﬁcance is explained fully in the next section. It guarantees that univariate elimination ideals for the constraint equations are generated by univariate polynomials. Un- fortunately, to maintain zero dimensionality our contraction scheme to the doublet must operate entirely in the framework of maximally independent **graphs**, and it is this that necessitates the extended graph theory of Section 4.

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Claim 4. For every z ∈ C, there is at most one blue **edge**, say zu, between z and A r and if this unique **edge** zu exists in G c , then there is no red **edge** zx in G c .
Proof. Assume by contradiction that there are at least **two** blue edges, say zu and zv in G c u, v ∈ A r . Consider a path P from x to z whose last **edge** is red. Clearly such a path exists in G c , by the definition of C. If u is not on this path, then u ∈ C, since P ∪ zu defines a path from x to u whose last **edge** is blue, a contradiction to the definitions of A r and C. Similar arguments hold if we consider v instead of u. Consequently, we conclude that both u and v belong to P . Let u − (respectively u + ) denote the predecessor (respectively the sucessor) of u, when we go from x to z along P . Analogously we define v − and v + and z − . As u and v are both vertices of A r , the edges u − u and v − v are both red. Furtehrmore, as P is a properly **edge**-colored path, it follows that both edges uu + and vv + are blue. Now by considering the path x · · · u − uzz − · · · v − v between x and v, we conclude that v ∈ C, a contradiction to the definitions of A r and C. This proves that there exists at most one **edge** between each vertex z ∈ C and A r . Remains to prove that if for some vertex z ∈ C, this unique **edge**, say zu, u ∈ A r exists in G c , then the **edge** zx (if any) is not a red one. Assume therefore that a red **edge** xz exists in G c . But then the path x − z r − u exists well in G b c and its last **edge** is a blue one. Thus u ∈ C, again a contradiction, since A r and C are vertex **disjoint** by definition. This completes the proof of the claim.

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=⇒ max flow value H is at least k.
Proof: Given k such **edge** **disjoint** paths, push one unit of flow along each such path. The resulting flow is legal in h and it has value k.
Definition 13.1.2 (0/1-flow). A flow f is a 0/1-flow if every **edge** has either no flow on it, or one unit of flow.

Abstract—The presence of **edge**-**disjoint** Hamiltonian cycles provides an advantage when implementing algorithms that require a ring structure by allowing message traffic to be spread evenly across the network. **Edge**-**disjoint** Hamiltonian cycles also provide the **edge**-fault tolerant Hamiltonicity of an interconnection network. **Two** node-**disjoint** cycles in a network are called equal if the number of nodes in the **two** cycles are the same and every node appears in one cycle exactly once. The presence of **two** equal node-**disjoint** cycles provides algorithms that require a ring structure to be preformed in the network simultaneously. The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. The n -dimensional twisted cube, an important variation of the hypercube, possesses some properties superior to the hypercube. In this paper, we present linear time algorithms to construct **two** **edge**-**disjoint** Hamiltonian cycles and **two** equal node-**disjoint** cycles in an n -dimensional twisted cube.

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Our algorithms do not provide any certificate for their correctness. This could be a Henneberg sequence [17] or a decomposition into **two** acyclic subgraphs [7]. We do not know how to compute either of these faster than using the algorithm of Gabow and Westermann [13]. Further our strategy heavily depends on planarity. Thus it seems unlikely that we can extend our approach to nonplanar **Laman** **graphs**. Instead it seems interesting to extend our strategies to more general pseudotriangulations. Orden et al. [29] show a characterization of general pseudotriangulations that extends the one for PPTs described in Section 4. Again there is a notion of combinatorial pseudotriangulations (CPT) which are stretchable if and only if certain combinatorial conditions hold. Besides a connectivity condition similar to the one for CPPTs there is an additional condition on sizes of subgraphs. In particular, a CPT might not be stretchable although the underlying graph is realizable as a pseudotriangulation. We do not know how to identify the stretchable CPTs.

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In computer science, **graphs** are used in variety of applications directly or indirectly. Especially quantitative labeled **graphs** have played a vital role in computational linguistics, decision making software tools, coding theory and path determination in networks. For a graph G(V, E) with the vertex set V and the **edge** set E, a vertex k-labeling φ : V → {1, 2, . . . , k} is defined to be an **edge** irregular k-labeling of the graph G if for every **two** different edges e and f their w φ (e) 6= w φ (f ), where the weight of an **edge** e = xy ∈ E(G) is w φ (xy) = φ(x) + φ(y). The minimum k for which the graph G has an **edge** irregular k-labeling is called the **edge** irregularity strength of G, denoted by es(G). In this paper, we determine the **edge** irregularity strengths of some chain **graphs** and the join of **two** **graphs**. We introduce a conjecture and open problems for researchers for further research.

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Sergiy Kozerenko
Communicated by V. V. Kirichenko
A b s t r a c t . Given a pair (X, σ) consisting of a finite tree X and its vertex self-map σ one can construct the corresponding Markov graph Γ(X, σ) which is a digraph that encodes σ-covering relation between edges in X . M-**graphs** are Markov **graphs** up to isomorphism. We obtain several sufficient conditions for the **disjoint** union of M-**graphs** to be an M-graph and prove that each weak component of M-graph is an M-graph itself.

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V G v v v and **edge** set E G ( ) = { , e e 1 2 , , e m } . We denote the shortest distance between **two** vertices u and v in G by d u v ( , ) and the degree of a vertex v in G by d v ( ) . A topological index is a real number derived from the structure of a graph, which is invariant under graph isomorphism. The Wiener index W(G) of a graph G is defined as

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Abstract. A fuzzy graph can be obtained from **two** given fuzzy **graphs** using union and join. In this paper, we find the degree of an **edge** in fuzzy **graphs** formed by these operations in terms of the degree of edges in the given fuzzy **graphs** in some particular cases.

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