On the geodetic number of permutation graphs

A vertex set S in a graph G is a geodetic set if every vertex of G lies on some u?v geodesic of G, where u,v∈S. The geodetic number g(G) of G is the minimum cardinality over all geodetic sets of G. Let G ₁ and G ₂ be disjoint copies of a graph G, and let σ:V(G ₁)→V(G ₂) be a bijection. Then, a permutation graph G _σ =(V,E) has the vertex set V=V(G ₁)∪V(G ₂) and the edge set E=E(G ₁)∪E(G ₂)∪{uv∣v=σ(u)}. For any connected graph G of order n≥3, we prove the sharp bounds 2≤g(G _σ )≤2n?(1+△(G)), where △(G) denotes the maximum degree of G. We give examples showing that neither is there a function h ₁ such that g(G)<h ₁(g(G _σ )) for all pairs (G,σ), nor is there a function h ₂ such that h ₂(g(G))>g(G _σ ) for all pairs (G,σ). Further, we characterize permutation graphs G _σ satisfying g(G _σ )=2|V(G)|?(1+△(G)) when G is a cycle, a tree, or a complete k-partite graph.     

Geodetic and Steiner geodetic sets in 3-Steiner distance hereditary graphs

Let G be a connected graph and S⊆V(G). Then the Steiner distance of S, denoted by d_G(S), is the smallest number of edges in a connected subgraph of G containing S. Such a subgraph is necessarily a tree called a Steiner tree for S. The Steiner interval for a set S of vertices in a graph, denoted by I(S) is the union of all vertices that belong to some Steiner tree for S. If S={u,v}, then I(S) is the interval I[u,v] between u and v. A connected graph G is 3-Steiner distance hereditary (3-SDH) if, for every connected induced subgraph H of order at least 3 and every set S of three vertices of H, d_H(S)=d_G(S). The eccentricity of a vertex v in a connected graph G is defined as e(v)=max{d(v,x)|x∈V(G)}. A vertex v in a graph G is a contour vertex if for every vertex u adjacent with v, e(u)?e(v). The closure of a set S of vertices, denoted by I[S], is defined to be the union of intervals between pairs of vertices of S taken over all pairs of vertices in S. A set of vertices of a graph G is a geodetic set if its closure is the vertex set of G. The smallest cardinality of a geodetic set of G is called the geodetic number of G and is denoted by g(G). A set S of vertices of a connected graph G is a Steiner geodetic set for G if I(S)=V(G). The smallest cardinality of a Steiner geodetic set of G is called the Steiner geodetic number of G and is denoted by sg(G). We show that the contour vertices of 3-SDH and HHD-free graphs are geodetic sets. For 3-SDH graphs we also show that g(G)?sg(G). An efficient algorithm for finding Steiner intervals in 3-SDH graphs is developed.     

The geodetic numbers of graphs and digraphs

For every two vertices u and v in a graph G,a u-v geodesic is a shortest path between u and v.Let I(u,v)denote the set of all vertices lying on a u-v geodesic.For a vertex subset S,let I(S) denote the union of all I(u,v)for u,v∈S.The geodetic number g(G)of a graph G is the minimum cardinality of a set S with I(S)=V(G).For a digraph D,there is analogous terminology for the geodetic number g(D).The geodetic spectrum of a graph G,denoted by S(G),is the set of geodetic numbers of all orientations of graph G.The lower geodetic number is g~-(G)=minS(G)and the upper geodetic number is g~ (G)=maxS(G).The main purpose of this paper is to study the relations among g(G),g~-(G)and g~ (G)for connected graphs G.In addition,a sufficient and necessary condition for the equality of g(G)and g(G×K_2)is presented,which improves a result of Chartrand,Harary and Zhang.     

A characterization of block graphs

A block graph is a graph whose blocks are cliques. For each edge e=uv of a graph G, let N_e(u) denote the set of all vertices in G which are closer to u than v. In this paper we prove that a graph G is a block graph if and only if it satisfies two conditions: (a) The shortest path between any two vertices of G is unique; and (b) For each edge e=uv∈E(G), if x∈N_e(u) and y∈N_e(v), then, and only then, the shortest path between x and y contains the edge e. This confirms a conjecture of Dobrynin and Gutman [A.A. Dobrynin, I. Gutman, On a graph invariant related to the sum of all distances in a graph, Publ. Inst. Math., Beograd. 56 (1994) 18-22].     

Steiner intervals and Steiner geodetic numbers in distance-hereditary graphs

A Steiner tree for a set S of vertices in a connected graph G is a connected subgraph of G with a smallest number of edges that contains S. The Steiner interval I(S) of S is the union of all the vertices of G that belong to some Steiner tree for S. If S={u,v}, then I(S)=I[u,v] is called the interval between u and v and consists of all vertices that lie on some shortest u-v path in G. The smallest cardinality of a set S of vertices such that ?_u,v∈SI[u,v]=V(G) is called the geodetic number and is denoted by g(G). The smallest cardinality of a set S of vertices of G such that I(S)=V(G) is called the Steiner geodetic number of G and is denoted by sg(G). We show that for distance-hereditary graphs g(G)?sg(G) but that g(G)/sg(G) can be arbitrarily large if G is not distance hereditary. An efficient algorithm for finding the Steiner interval for a set of vertices in a distance-hereditary graph is described and it is shown how contour vertices can be used in developing an efficient algorithm for finding the Steiner geodetic number of a distance-hereditary graph.     

The upper connected geodetic number and forcing connected geodetic number of a graph

For a connected graph G of order p≥2, a set S⊆V(G) is a geodetic set of G if each vertex v∈V(G) lies on an x-y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set of G is defined as the geodetic number of G, denoted by g(G). A geodetic set of cardinality g(G) is called a g-set of G. A connected geodetic set of G is a geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected geodetic set of G is the connected geodetic number of G and is denoted by g_c(G). A connected geodetic set of cardinality g_c(G) is called a g_c-set of G. A connected geodetic set S in a connected graph G is called a minimal connected geodetic set if no proper subset of S is a connected geodetic set of G. The upper connected geodetic number is the maximum cardinality of a minimal connected geodetic set of G. We determine bounds for and determine the same for some special classes of graphs. For positive integers r,d and n≥d+1 with r≤d≤2r, there exists a connected graph G with , and . Also, for any positive integers 2≤a<b≤c, there exists a connected graph G such that g(G)=a, g_c(G)=b and . A subset T of a g_c-set S is called a forcing subset for S if S is the unique g_c-set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing connected geodetic number of S, denoted by f_c(S), is the cardinality of a minimum forcing subset of S. The forcing connected geodetic number of G, denoted by f_c(G), is f_c(G)=min{f_c(S)}, where the minimum is taken over all g_c-sets S in G. It is shown that for every pair a,b of integers with 0≤a≤b−4, there exists a connected graph G such that f_c(G)=a and g_c(G)=b.     

The forcing hull and forcing geodetic numbers of graphs

For every pair of vertices u,v in a graph, a u-v geodesic is a shortest path from u to v. For a graph G, let I_G[u,v] denote the set of all vertices lying on a u-v geodesic. Let S⊆V(G) and I_G[S] denote the union of all I_G[u,v] for all u,v∈S. A subset S⊆V(G) is a convex set of G if I_G[S]=S. A convex hull [S]_G of S is a minimum convex set containing S. A subset S of V(G) is a hull set of G if [S]_G=V(G). The hull number h(G) of a graph G is the minimum cardinality of a hull set in G. A subset S of V(G) is a geodetic set if I_G[S]=V(G). The geodetic number g(G) of a graph G is the minimum cardinality of a geodetic set in G. A subset F⊆V(G) is called a forcing hull (or geodetic) subset of G if there exists a unique minimum hull (or geodetic) set containing F. The cardinality of a minimum forcing hull subset in G is called the forcing hull number f_h(G) of G and the cardinality of a minimum forcing geodetic subset in G is called the forcing geodetic number f_g(G) of G. In the paper, we construct some 2-connected graph G with (f_h(G),f_g(G))=(0,0),(1,0), or (0,1), and prove that, for any nonnegative integers a, b, and c with a+b≥2, there exists a 2-connected graph G with (f_h(G),f_g(G),h(G),g(G))=(a,b,a+b+c,a+2b+c) or (a,2a+b,a+b+c,2a+2b+c). These results confirm a conjecture of Chartrand and Zhang proposed in [G. Chartrand, P. Zhang, The forcing hull number of a graph, J. Combin. Math. Combin. Comput. 36 (2001) 81-94].     

On global domination critical graphs

A dominating set of a graph G=(V,E) is a subset S⊆V such that every vertex not in S is adjacent to at least one vertex of S. The domination number of G is the cardinality of a smallest dominating set. The global domination number, γ_g(G), is the cardinality of a smallest set S that is simultaneously a dominating set of both G and its complement . Graphs for which γ_g(G−e)>γ_g(G) for all edges e∈E are characterized, as are graphs for which γ_g(G−e)<γ_g(G) for all edges e∈E whenever is disconnected. Progress is reported in the latter case when is connected.     

Competitively Tight Graphs

The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is the smallest number of such isolated vertices. Computing the competition number of a graph is an NP-hard problem in general and has been one of the important research problems in the study of competition graphs. Opsut [1982] showed that the competition number of a graph G is related to the edge clique cover number θ _E (G) of the graph G via θ _E (G) ? |V(G)| + 2 ≤ k(G) ≤ θ _E (G). We first show that for any positive integer m satisfying 2 ≤ m ≤ |V(G)|, there exists a graph G with k(G) = θ _E (G) ? |V(G)| + m and characterize a graph G satisfying k(G) = θ _E (G). We then focus on what we call competitively tight graphs G which satisfy the lower bound, i.e., k(G) = θ _E (G) ? |V(G)| + 2. We completely characterize the competitively tight graphs having at most two triangles. In addition, we provide a new upper bound for the competition number of a graph from which we derive a sufficient condition and a necessary condition for a graph to be competitively tight.     

Signed star domatic number of a graph

Let G be a simple graph without isolated vertices with vertex set V(G) and edge set E(G). A function f:E(G)?{−1,1} is said to be a signed star dominating function on G if ∑_e∈E(v)f(e)≥1 for every vertex v of G, where E(v)={uv∈E(G)∣u∈N(v)}. A set {f₁,f₂,…,f_d} of signed star dominating functions on G with the property that for each e∈E(G), is called a signed star dominating family (of functions) on G. The maximum number of functions in a signed star dominating family on G is the signed star domatic number of G, denoted by d_SS(G).In this paper we study the properties of the signed star domatic number d_SS(G). In particular, we determine the signed domatic number of some classes of graphs.     

The geodetic number of the lexicographic product of graphs

A set S of vertices of a graph G is a geodetic set if every vertex of G lies in an interval between two vertices from S. The size of a minimum geodetic set in G is the geodetic number g(G) of G. We find that the geodetic number of the lexicographic product G°H for a non-complete graph H lies between 2 and 3g(G). We characterize the graphs G and H for which g(G°H)=2, as well as the lexicographic products T°H that enjoy g(T°H)=3g(G), when T is isomorphic to a tree. Using a new concept of the so-called geodominating triple of a graph G, a formula that expresses the exact geodetic number of G°H is established, where G is an arbitrary graph and H a non-complete graph.     

Properties of total domination edge-critical graphs

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 γ_t(G) of G. The graph G is total domination edge critical if for every edge e in the complement of G, γ_t(G+e)<γ_t(G). We call such graphs γ_tEC. Properties of γ_tEC graphs are established.     

The hull and geodetic numbers of orientations of graphs

For an oriented graph D, let I_D[u,v] denote the set of all vertices lying on a u-v geodesic or a v-u geodesic. For S⊆V(D), let I_D[S] denote the union of all I_D[u,v] for all u,v∈S. Let [S]_D denote the smallest convex set containing S. The geodetic number g(D) of an oriented graph D is the minimum cardinality of a set S with I_D[S]=V(D) and the hull number h(D) of an oriented graph D is the minimum cardinality of a set S with [S]_D=V(D). For a connected graph G, let O(G) be the set of all orientations of G, define g⁻(G)=min{g(D):D∈O(G)}, g⁺(G)=max{g(D):D∈O(G)}, h⁻(G)=min{h(D):D∈O(G)}, and h⁺(G)=max{h(D):D∈O(G)}. By the above definitions, h⁻(G)≤g⁻(G) and h⁺(G)≤g⁺(G). In the paper, we prove that g⁻(G)<h⁺(G) for a connected graph G of order at least 3, and for any nonnegative integers a and b, there exists a connected graph G such that g⁻(G)−h⁻(G)=a and g⁺(G)−h⁺(G)=b. These results answer a problem of Farrugia in [A. Farrugia, Orientable convexity, geodetic and hull numbers in graphs, Discrete Appl. Math. 148 (2005) 256-262].     

Representing Graphs in Steiner Triple Systems

Let G =  (V, E) be a simple graph and let T =  (P, B) be a Steiner triple system. Let φ be a one-to-one function from V to P. Any edge e =  {u, v} has its image {φ(u), φ(v)} in a unique block in B. We also denote this induced function from edges to blocks by φ. We say that T represents G if there exists a one-to-one function φ : V → P such that the induced function φ : E → B is also one-to-one; that is, if we can represent vertices of the graph by points of the triple system such that no two edges are represented by the same block. In this paper we examine when a graph can be represented by an STS. First, we find a bound which ensures that every graph of order n is represented in some STS of order f(n). Second, we find a bound which ensures that every graph of order n is represented in every STS of order g(n). Both of these answers are related to finding an independent set in an STS. Our question is a generalization of finding such independent sets. We next examine which graphs can be represented in STS’s of small orders. Finally, we give bounds on the orders of STS’s that are guaranteed to embed all graphs of a given maximum degree.     

On the spectral spread of bicyclic graphs with given girth

The spectral spread of a graph is defined to be the difference between the largest and the least eigenvalue of the adjacency matrix of the graph. A graph G is said to be bicyclic, if G is connected and |E(G)| = |V(G)|+ 1. Let B(n, g) be the set of bicyclic graphs on n vertices with girth g. In this paper some properties about the least eigenvalues of graphs are given, by which the unique graph with maximal spectral spread in B(n, g) is determined.     

The Chromatic Index of a Graph Whose Core is a Cycle of Order at Most 13

Let G be a graph. The core of G, denoted by G _Δ, is the subgraph of G induced by the vertices of degree Δ(G), where Δ(G) denotes the maximum degree of G. A k -edge coloring of G is a function f : E(G) → L such that |L| = k and f (e ₁) ≠ f (e ₂) for all two adjacent edges e ₁ and e ₂ of G. The chromatic index of G, denoted by χ′(G), is the minimum number k for which G has a k-edge coloring. A graph G is said to be Class 1 if χ′(G) = Δ(G) and Class 2 if χ′(G) = Δ(G) + 1. In this paper it is shown that every connected graph G of even order whose core is a cycle of order at most 13 is Class 1.     

Minimally 3-restricted edge connected graphs

For a connected graph G=(V,E), an edge set S⊂E is a 3-restricted edge cut if G−S is disconnected and every component of G−S has order at least three. The cardinality of a minimum 3-restricted edge cut of G is the 3-restricted edge connectivity of G, denoted by λ₃(G). A graph G is called minimally 3-restricted edge connected if λ₃(G−e)<λ₃(G) for each edge e∈E. A graph G is λ₃-optimal if λ₃(G)=ξ₃(G), where , ω(U) is the number of edges between U and V?U, and G[U] is the subgraph of G induced by vertex set U. We show in this paper that a minimally 3-restricted edge connected graph is always λ₃-optimal except the 3-cube.     

On the remoteness function in median graphs

A profile on a graph G is any nonempty multiset whose elements are vertices from G. The corresponding remoteness function associates to each vertex x∈V(G) the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O(mlogn) time whether G is a median graph with geodetic number 2.