Total domination critical and stable graphs upon edge removal

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. A graph is total domination edge critical if the removal of any arbitrary edge increases the total domination number. On the other hand, a graph is total domination edge stable if the removal of any arbitrary edge has no effect on the total domination number. In this paper, we characterize total domination edge critical graphs. We also investigate various properties of total domination edge stable graphs.     

Edge-connectivity and edge-superconnectivity in sequence graphs

Given an integer k?1 and any graph G, the sequence graph S_k(G) is the graph whose set of vertices is the set of all walks of length k in G. Moreover, two vertices of S_k(G) are joined by an edge if and only if their corresponding walks are adjacent in G.In this paper we prove sufficient conditions for a sequence graph S_k(G) to be maximally edge-connected and edge-superconnected depending on the parity of k and on the vertex-connectivity of the original graph G.     

A note on the computational complexity of graph vertex partition

A stable set of a graph is a vertex set in which any two vertices are not adjacent. It was proven in [A. Brandstädt, V.B. Le, T. Szymczak, The complexity of some problems related to graph 3-colorability, Discrete Appl. Math. 89 (1998) 59-73] that the following problem is NP-complete: Given a bipartite graph G, check whether G has a stable set S such thatG-Sis a tree. In this paper we prove the following problem is polynomially solvable: Given a graph G with maximum degree 3 and containing no vertices of degree 2, check whether G has a stable set S such thatG-Sis a tree. Thus we partly answer a question posed by the authors in the above paper. Moreover, we give some structural characterizations for a graph G with maximum degree 3 that has a stable set S such that G-S is a tree.     

Some results on exclusive sum graphs

Let N(Z) denote the set of all positive integers (integers). The sum graph G ⁺(S) of a finite subset S?N(Z) is the graph (S,E) with uv∈E if and only if u+v∈S. A graph G is said to be an (integral) sum graph if it is isomorphic to the sum graph of some S?N(Z). A sum labelling S is called an exclusive sum labelling if u+v∈S?V(G) for any edge uv∈E(G). We say that G is labeled exclusively. The least number r of isolated vertices such that G∪rK ₁ is an exclusive sum graph is called the exclusive sum number ε(G) of graph G. In this paper, we discuss the exclusive sum number of disjoint union of two graphs and the exclusive sum number of some graph classes.     

Clique-inserted-graphs and spectral dynamics of clique-inserting

Motivated by studying the spectra of truncated polyhedra, we consider the clique-inserted-graphs. For a regular graph G of degree r>0, the graph obtained by replacing every vertex of G with a complete graph of order r is called the clique-inserted-graph of G, denoted as C(G). We obtain a formula for the characteristic polynomial of C(G) in terms of the characteristic polynomial of G. Furthermore, we analyze the spectral dynamics of iterations of clique-inserting on a regular graph G. For any r-regular graph G with r>2, let S(G) denote the union of the eigenvalue sets of all iterated clique-inserted-graphs of G. We discover that the set of limit points of S(G) is a fractal with the maximum r and the minimum −2, and that the fractal is independent of the structure of the concerned regular graph G as long as the degree r of G is fixed. It follows that for any integer r>2 there exist infinitely many connected r-regular graphs (or, non-regular graphs with r as the maximum degree) with arbitrarily many distinct eigenvalues in an arbitrarily small interval around any given point in the fractal. We also present a formula on the number of spanning trees of any kth iterated clique-inserted-graph and other related results.     

Whiskers and sequentially Cohen-Macaulay graphs

We investigate how to modify a simple graph G combinatorially to obtain a sequentially Cohen-Macaulay graph. We focus on adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex of G. We give various sufficient conditions and necessary conditions on a subset S of the vertices of G so that the graph G∪W(S), obtained from G by adding a whisker to each vertex in S, is a sequentially Cohen-Macaulay graph. For instance, we show that if S is a vertex cover of G, then G∪W(S) is a sequentially Cohen-Macaulay graph. On the other hand, we show that if G?S is not sequentially Cohen-Macaulay, then G∪W(S) is not a sequentially Cohen-Macaulay graph. Our work is inspired by and generalizes a result of Villarreal on the use of whiskers to get Cohen-Macaulay graphs.     

Some results related to the toughness of 3-domination critical graphs

A graph G is said to be k-γ-critical if the size of any minimum dominating set of vertices is k, but if any edge is added to G the resulting graph can be dominated with k−1 vertices. The structure of k-γ-critical graphs remains far from completely understood, even in the special case when the domination number γ=3. In a 1983 paper, Sumner and Blitch proved a theorem which may regarded as a result related to the toughness of 3-γ-critical graphs which says that if S is any vertex cutset of such a graph, then G−S has at most |S|+1 components. In the present paper, we improve and extend this result considerably.     

On the matching polynomial of subdivision graphs

Let G be a simple graph and let S(G) be the subdivision graph of G, which is obtained from G by replacing each edge of G by a path of length two. In this paper, by the Principle of Inclusion and Exclusion we express the matching polynomial and Hosoya index of S(G) in terms of the matchings of G. Particularly, if G is a regular graph or a semi-regular bipartite graph, then the closed formulae of the matching polynomial and Hosoya index of S(G) are obtained. As an application, we prove a combinatorial identity.     

The inverse inertia problem for graphs: Cut vertices, trees, and a counterexample

Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which inertias can be attained by a matrix in S(G). We give a complete answer to this question for trees in terms of a new family of graph parameters, the maximal disconnection numbers of a graph. We also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs. Finally, we give an example of a graph that is not inertia-balanced, which settles an open problem from the October 2006 AIM Workshop on Spectra of Families of Matrices described by Graphs, Digraphs and Sign Patterns. We also determine some restrictions on the inertia set of any graph.     

Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids

A matching M is uniquely restricted in a graph G if its saturated vertices induce a subgraph which has a unique perfect matching, namely M itself [M.C. Golumbic, T. Hirst, M. Lewenstein, Uniquely restricted matchings, Algorithmica 31 (2001) 139-154]. G is a König-Egerváry graph provided α(G)+μ(G)=|V(G)| [R.W. Deming, Independence numbers of graphs—an extension of the König-Egerváry theorem, Discrete Math. 27 (1979) 23-33; F. Sterboul, A characterization of the graphs in which the transversal number equals the matching number, J. Combin. Theory Ser. B 27 (1979) 228-229], where μ(G) is the size of a maximum matching and α(G) is the cardinality of a maximum stable set. S is a local maximum stable set of G, and we write S∈Ψ(G), if S is a maximum stable set of the subgraph spanned by S∪N(S), where N(S) is the neighborhood of S. Nemhauser and Trotter [Vertex packings: structural properties and algorithms, Math. Programming 8 (1975) 232-248], proved that any S∈Ψ(G) is a subset of a maximum stable set of G. In [V.E. Levit, E. Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math. 132 (2003) 163-174] we have proved that for a bipartite graph G,Ψ(G) is a greedoid on its vertex set if and only if all its maximum matchings are uniquely restricted. In this paper we demonstrate that if G is a triangle-free graph, then Ψ(G) is a greedoid if and only if all its maximum matchings are uniquely restricted and for any S∈Ψ(G), the subgraph spanned by S∪N(S) is a König-Egerváry graph.     

Edge covering coloring of nearly bipartite graphs

LetG be a simple graph with vertex setV(G) and edge setE(G). A subsetS ofE(G) is called an edge cover ofG if the subgraph induced byS is a spanning subgraph ofG. The maximum number of edge covers which form a partition ofE(G) is called edge covering chromatic number ofG, denoted by χ′_c(G). It known that for any graphG with minimum degreeδ,δ -1 ≤χ′_c(G) ≤δ. If χ′_c(G) =δ, thenG is called a graph of CI class, otherwiseG is called a graph of CII class. It is easy to prove that the problem of deciding whether a given graph is of CI class or CII class is NP-complete. In this paper, we consider the classification of nearly bipartite graph and give some sufficient conditions for a nearly bipartite graph to be of CI class.     

Total domination dot-stable graphs

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. Two vertices of G are said to be dotted (identified) if they are combined to form one vertex whose open neighborhood is the union of their neighborhoods minus themselves. We note that dotting any pair of vertices cannot increase the total domination number. Further we show it can decrease the total domination number by at most 2. A graph is total domination dot-stable if dotting any pair of adjacent vertices leaves the total domination number unchanged. We characterize the total domination dot-stable graphs and give a sharp upper bound on their total domination number. We also characterize the graphs attaining this bound.     

On the bondage number of middle graphs

Let G = (V (G),E(G)) be a simple graph. A subset S of V (G) is a dominating set of G if, for any vertex v ∈ V (G) — S, there exists some vertex u ∈ S such that uv ∈ E(G). The domination number, denoted by γ(G), is the cardinality of a minimal dominating set of G. There are several types of domination parameters depending upon the nature of domination and the nature of dominating set. These parameters are bondage, reinforcement, strong-weak domination, strong-weak bondage numbers. In this paper, we first investigate the strong-weak domination number of middle graphs of a graph. Then several results for the bondage, strong-weak bondage number of middle graphs are obtained.     

Toughness and the existence of fractional k-factors of graphs

The toughness of a graph G, t(G), is defined as t(G)=min{|S|/ω(G-S)|S⊆V(G),ω(G-S)>1} where ω(G-S) denotes the number of components of G-S or t(G)=+∞ if G is a complete graph. Much work has been contributed to the relations between toughness and the existence of factors of a graph. In this paper, we consider the relationship between the toughness and the existence of fractional k-factors. It is proved that a graph G has a fractional 1-factor if t(G)?1 and has a fractional k-factor if t(G)?k-1/k where k?2. Furthermore, we show that both results are best possible in some sense.     

Factor-critical property in 3-dominating-critical graphs

A vertex subset S of a graph G is a dominating set if every vertex of G either belongs to S or is adjacent to a vertex of S. The cardinality of a smallest dominating set is called the dominating number of G and is denoted by γ(G). A graph G is said to be γ-vertex-critical if γ(G−v)<γ(G), for every vertex v in G.Let G be a 2-connected K_1,5-free 3-vertex-critical graph of odd order. For any vertex v∈V(G), we show that G−v has a perfect matching (except two graphs), which solves a conjecture posed by Ananchuen and Plummer [N. Ananchuen, M.D. Plummer, Matchings in 3-vertex critical graphs: The odd case, Discrete Math., 307 (2007) 1651-1658].     

Random mappings of scaled graphs

Given a connected finite graph Γ with a fixed base point O and some graph G with a based point we study random 1-Lipschitz maps of a scaled Γ into G. We are mostly interested in the case where G is a Cayley graph of some finitely generated group, where the construction does not depend on the choice of base points. A particular case of Γ being a graph on two vertices and one edge corresponds to the random walk on G, and the case where Γ is a graph on two vertices and two edges joining them corresponds to Brownian bridge in G. We show, that unlike in the case ${G=\mathbb Z^d}$ , the asymptotic behavior of a random scaled mapping of Γ into G may differ significantly from the asymptotic behavior of random walks or random loops in G. In particular, we show that this occurs when G is a free non-Abelian group. Also we consider the case when G is a wreath product of ${\mathbb Z}$ with a finite group. To treat this case we prove new estimates for transition probabilities in such wreath products. For any group G generated by a finite set S we define a functor E from category of finite connected graphs to the category of equivalence relations on such graphs. Given a finite connected graph Γ, the value E _G,S (Γ) can be viewed as an asymptotic invariant of G.     

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ^?1. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ^?1 and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

Maximum nullity of outerplanar graphs and the path cover number

Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[a_ij] with a_ij≠0,i≠j if and only if ij∈E. By M(G) we denote the largest possible nullity of any matrix A∈S(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G).     

A sharp upper bound on algebraic connectivity using domination number

Let G be a connected graph of order n. The algebraic connectivity of G is the second smallest eigenvalue of the Laplacian matrix of G. A dominating set in G is a vertex subset S such that each vertex of G that is not in S is adjacent to a vertex in S. The least cardinality of a dominating set is the domination number. In this paper, we prove a sharp upper bound on the algebraic connectivity of a connected graph in terms of the domination number and characterize the associated extremal graphs.     

Structure of vertex-transitive graphs

Every vertex-transitive graph has a characteristic structure. The specific details of structure of a vertex-transitive graph are closely related to the configuration of the parts of any minimum separating set of that graph. It is shown for a vertex-transitive graph G that (i) the number n of isomorphic atomic parts admitted by any minimum separating set S is unique for that G; (ii) G is then isomorphic to a disjoint union of m(≥2) copies of such a set of n atomic parts together with some additional edges joining them; (iii) any minimum separating set S of G consists of the vertex set of the union of some k(≥1) copies of the set of n atomic parts; (iv) at most one nonatomic part will be admitted in conjunction with one or more atomic parts by any minimum separating set of G.This configuration of structure extends to nonatomic parts. A vertex-transitive graph G may contain a not necessarily unique minimum separating set S which admits t(≥3) nonatomic parts. Then it is shown that (i) some (t ? 1)-set of these nonatomic parts are pairwise isomorphic; (ii) if the remaining nonatomic part is nonisomorphic to the others then it contains more vertices than the other parts; (iii) G is isomorphic to a disjoint union of d(≥2) copies of the set of q(≥t ? 1) isomorphic nonatomic parts together with some additional edges joining them; (iv) a minimum separating set S consists of the vertex set of the union of some y(≥1) copies of the set of q isomorphic nonatomic parts. For the case of t = 2 non-atomic parts admitted by a minimum separating set S, counterexamples of (iii) and (iv) are given.