On a graph’s security number

A secure set S⊆V of graph G=(V,E) is a set whose every nonempty subset can be successfully defended from an attack, under appropriate definitions of “attack” and “defended.” The set S is secure when |N[X]∩S|≥|N[X]−S| for every X⊆S. The smallest cardinality of a secure set in G is the security number of G. New bounds for the security number are established, the effect of some graph modifications on the security number is investigated, and the exact value of the security number for some families of graphs is given.     

Strong matching preclusion under the conditional fault model

Strong matching preclusion that additionally permits more destructive vertex faults in a graph [J.-H. Park, I. Ihm, Strong matching preclusion, Theoretical Computer Science 412 (2011) 6409–6419] is an extended form of the original matching preclusion that assumes only edge faults [R.C. Brigham, F. Harary, E.C. Violin, J. Yellen, Perfect-matching preclusion, Congressus Numerantium 174 (2005) 185–192]. In this paper, we study the problem of strong matching preclusion under the condition that no isolated vertex is created as a result of faults. After briefly discussing some fundamental classes of graphs in the point of the conditional matching preclusion, we establish the conditional strong matching preclusion number for the class of restricted hypercube-like graphs, which include most nonbipartite hypercube-like networks found in the literature.     

The security number of lexicographic products

A subset S of vertices of a graph G is a secure set if |N [X] ∩ S| ≥ |N [X] ? S| holds for any subset X of S, where N [X] denotes the closed neighborhood of X. The minimum cardinality s(G) of a secure set in G is called the security number of G. We investigate the security number of lexicographic product graphs by defining a new concept of tightly-securable graphs. In particular we derive several exact results for different families of graphs which yield some general results.     

A lower bound for the independence number of a planar graph

A subset of the vertices of a graph is independent if no two vertices in the subset are adjacent. The independence number α(G) is the maximum number of vertices in an independent set. We prove that if G is a planar graph on N vertices then $\text{α(G)/N ?}\text{2}\text{9}$.     

Induced embeddings in Steinhaus graphs

Fix any positive integer n. Let S be the set of all Steinhaus graphs of order n(n − 1)/2 + 1. The vertices for each graph in S are the first n(n − 1)/2 + 1 positive integers. Let I be the set of all labeled graphs of order n with vertices of the form i(i − 1)/2 + 1 for the first n positive integers i. This article shows that the function ϕ : S → I that maps a Steinhaus graph to its induced subgraph is a bijection. Therefore, any graph of order n is isomorphic to an induced subgraph of a Steinhaus graph of order n(n − 1)/2 + 1. This considerably tightens a result of Brigham, Carrington, and Dutton in [Brigham, Carrington, & Dutton, Combin. Inform. System Sci. 17 (1992)], which showed that this could be done with a Steinhaus graph of order 2ⁿ⁻¹. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 1–9, 1998     

The Rainbow Vertex-disconnection in Graphs

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set of G is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S of G such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of G-S; whereas when x and y are adjacent, S + x or S + y is rainbow and x and y belong to different components of(G-xy)-S. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by rvd(G), is the minimum number of colors that are needed to make G rainbow vertexdisconnected. In this paper, we characterize all graphs of order n with rainbow vertex-disconnection number k for k ∈ {1, 2, n}, and determine the rainbow vertex-disconnection numbers of some special graphs. Moreover, we study the extremal problems on the number of edges of a connected graph G with order n and rvd(G) = k for given integers k and n with 1 ≤ k ≤ n.     

A note on domination, girth and minimum degree

Let G be a graph of order n, minimum degree δ?2, girth g?5 and domination number γ. In 1990 Brigham and Dutton [Bounds on the domination number of a graph, Q. J. Math., Oxf. II. Ser. 41 (1990) 269-275] proved that γ?⌈n/2-g/6⌉. This result was recently improved by Volkmann [Upper bounds on the domination number of a graph in terms of diameter and girth, J. Combin. Math. Combin. Comput. 52 (2005) 131-141; An upper bound for the domination number of a graph in terms of order and girth, J. Combin. Math. Combin. Comput. 54 (2005) 195-212] who for i∈{1,2} determined a finite set of graphs G_i such that γ?⌈n/2-g/6-(3i+3)/6⌉ unless G is a cycle or G∈G_i.Our main result is that for every i∈N there is a finite set of graphs G_i such that γ?n/2-g/6-i unless G is a cycle or G∈G_i. Furthermore, we conjecture another improvement of Brigham and Dutton's bound and prove a weakened version of this conjecture.     

The structure of graphs with given lengths of cycles

The cycle spectrum of a given graph G is the lengths of cycles in G. In this paper, we introduce the following problem: determining the maximum number of edges of an n-vertex graph with given cycle spectrum. In particular, we determine the maximum number of edges of an n-vertex graph without containing cycles of lengths three and at least k. This can be viewed as an extension of a well-known result of Erd?s and Gallai concerning the maximum number of edges of an n-vertex graph without containing cycles of lengths at least k. We also determine the maximum number of edges of an n-vertex graph whose cycle spectrum is a subset of two positive integers.     

Total restrained bondage in graphs

A subset S of vertices of a graph G with no isolated vertex is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex in V (G) S is also adjacent to a vertex in V (G) S. The total restrained domination number of G is the minimum cardinality of a total restrained dominating set of G. In this paper we initiate the study of total restrained bondage in graphs. The total restrained bondage number in a graph G with no isolated vertex, is the minimum cardinality of a subset of edges E such that G E has no isolated vertex and the total restrained domination number of G E is greater than the total restrained domination number of G. We obtain several properties, exact values and bounds for the total restrained bondage number of a graph.     

和图、整和图与模和图的几个结果

Let N denote the set of positive integers. The sum graph G^＋（S） of a finite subset S belong to N is the graph （S, E） with uv ∈ E if and only if u ＋ v ∈ S. A graph G is said to be a sum graph if it is isomorphic to the sum graph of some S belong to N. By using the set Z of all integers instead of N, we obtain the definition of the integral sum graph. A graph G = （V, E） is a mod sum graph if there exists a positive integer z and a labelling, λ, of the vertices of G with distinct elements from {0, 1, 2,..., z - 1} so that uv ∈ E if and only if the sum, modulo z, of the labels assigned to u and v is the label of a vertex of G. In this paper, we prove that flower tree is integral sum graph. We prove that Dutch m-wind-mill （Dm） is integral sum graph and mod sum graph, and give the sum number of Dm.     

笛卡尔乘积图里的最小k-路点覆盖

对于子集$S\subseteq V(G)$,如果图$G$里的每一条$k$路都至少包含$S$中的一个点,那么我们称集合$S$是图$G$的一个$k$-路点覆盖.很明显,这个子集并不唯一.我们称最小的$k$-路点覆盖的基数为$k$-路点覆盖数, 记作$\psi_k(G)$.本文给出了一些笛卡尔乘积图上$\psi_k(G)$值的上界或下界.     

A polyhedral study of the maximum edge subgraph problem

The study of cohesive subgroups is an important aspect of social network analysis. Cohesive subgroups are studied using different relaxations of the notion of clique in a graph. For instance, given a graph and an integer k, the maximum edge subgraph problem consists in finding a k-vertex subset such that the number of edges within the subset is maximum. This work proposes a polyhedral approach for this NP-hard problem. We study the polytope associated to an integer programming formulation of the problem, present several families of facet-inducing valid inequalities, and discuss the separation problem associated to these families.     

Complexity results related to monophonic convexity

The study of monophonic convexity is based on the family of induced paths of a graph. The closure of a subset X of vertices, in this case, contains every vertex v such that v belongs to some induced path linking two vertices of X. Such a closure is called monophonic closure. Likewise, the convex hull of a subset is called monophonic convex hull. In this work we deal with the computational complexity of determining important convexity parameters, considered in the context of monophonic convexity. Given a graph G, we focus on three parameters: the size of a maximum proper convex subset of G (m-convexity number); the size of a minimum subset whose closure is equal to V(G) (monophonic number); and the size of a minimum subset whose convex hull is equal to V(G) (m-hull number). We prove that the decision problems corresponding to the m-convexity and monophonic numbers are NP-complete, and we describe a polynomial time algorithm for computing the m-hull number of an arbitrary graph.     

Some results on integral sum graphs

Let Z denote the set of all integers. The integral sum graph of a finite subset S of Z is the graph (S,E) with vertex set S and edge set E such that for u,vS, uvE if and only if u+vS. A graph G is called an integral sum graph if it is isomorphic to the integral sum graph of some finite subset S of Z. The integral sum number of a given graph G, denoted by ζ(G), is the smallest number of isolated vertices which when added to G result in an integral sum graph. Let x denote the least integer not less than the real x. In this paper, we (i) determine the value of ζ(K_n−E(K_r)) for r2n/3−1, (ii) obtain a lower bound for ζ(K_n−E(K_r)) when 2r<2n/3−1 and n5, showing by construction that the bound is sharp when r=2, and (iii) determine the value of ζ(K_r,r) for r2. These results provide partial solutions to two problems posed by Harary (Discrete Math. 124 (1994) 101–108). Finally, we furnish a counterexample to a result on the sum number of K_r,s given by Hartsfiedl and Smyth (Graphs and Matrices, R. Rees (Ed.), Marcel, Dekker, New York, 1992, pp. 205–211).     

Harary's conjectures on integral sum graphs

Let N denote the set of positive integers and Z denote all integers. The (integral) sum graph of a finite subset S N(Z) is the graph (S, E) with uv ε E if and only if u + v ε S. A graph G is said to be an (integral) sum graph if it is isomorphic to the (integral) sum graph of some S N(Z). The (integral) sum number of a given graph G is the smallest number of isolated nodes which when added to G result in an (integral) sum graph.

We show that the integral sum number of a complete graph with n 4 nodes equals 2n − 3, which proves a conjecture of Harary. And we disprove another conjecture of Harary by showing that there are infinitely many trees which are not caterpillars but are integral sum graphs.     

Strong chromatic index of subset graphs

The strong chromatic index of a graph G, denoted sq(G), is the minimum number of parts needed to partition the edges of G into induced matchings. For 0 ≤ k ≤ l ≤ m, the subset graph S_m(k, l) is a bipartite graph whose vertices are the k- and l-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. We show that and that this number satisfies the strong chromatic index conjecture by Brualdi and Quinn for bipartite graphs. Further, we demonstrate that the conjecture is also valid for a more general family of bipartite graphs. © 1997 John Wiley & Sons, Inc.     

Quasi-flats in semihyperbolic groups

We prove that the Cayley graph of a group which is semihyperbolic but not hyperbolic contains a subset quasi-isometric to .

     

The lower and upper forcing geodetic numbers of block–cactus graphs

A vertex set D in graph G is called a geodetic set if all vertices of G are lying on some shortest u–v path of G, where u, v ∈ D. The geodetic number of a graph G is the minimum cardinality among all geodetic sets. A subset S of a geodetic set D is called a forcing subset of D if D is the unique geodetic set containing S. The forcing geodetic number of D is the minimum cardinality of a forcing subset of D, and the lower and the upper forcing geodetic numbers of a graph G are the minimum and the maximum forcing geodetic numbers, respectively, among all minimum geodetic sets of G. In this paper, we find out the lower and the upper forcing geodetic numbers of block–cactus graphs.     

Maximum weight archipelago subgraph problem

This paper is devoted to a new problem of combinatorial optimization. The problem is called Maximum Weight Archipelago Subgraph Problem (MWASP). Archipelago is a signed graph such that the negative edges connect the components of the graph of the positive edges. The new problem is to find a subset of edges in a weighted signed graph such that (i) if the edges of the subset are deleted from the graph then the remaining graph is an archipelago; and (ii) the subset has minimal total weight among the subsets having property (i). The problem is NP-complete, however a polynomial algorithm is provided to obtain the maximal weight of an edge what is still necessary to delete. The problem MWAP is used to analyze the relation of the blue chips of the Dow Jones Index.     

A study on oriented relative clique number

An oriented graph is a directed graph with no directed cycle of length one or two. The relative clique number of an oriented graph is the cardinality of a largest subset $X$ of vertices such that each pair of vertices is either adjacent or connected by a directed 2-path. It is known that the oriented relative clique number of a planar graph is at most 80. Here we improve the upper bound to 32. We also prove an upper bound of 14 for oriented relative clique number of triangle-free planar graphs. Furthermore, we determine the exact values of oriented relative clique number for the families of outerplanar graphs with girth at least $g$ and planar graphs with girth at least $g+2$ for all $g\ge 3$. Moreover, we study the relation of oriented relative clique number with oriented chromatic number, oriented absolute clique number and maximum degree of a graph. We also show that oriented relative clique number of a connected subcubic graph is at most seven which weakly supports a conjecture by Sopena (JGT 1997).