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 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].     

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].     

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.     

Extreme Geodesic Graphs

For two vertices u and v of a graph G, the closed interval I[u, v] consists of u, v, and all vertices lying in some u–v geodesic of G, while for S V(G), the set I[S] is the union of all sets I[u, v] for u, v S. A set S of vertices of G for which I[S] = V(G) is a geodetic set for G, and the minimum cardinality of a geodetic set is the geodetic number g(G). A vertex v in G is an extreme vertex if the subgraph induced by its neighborhood is complete. The number of extreme vertices in G is its extreme order ex(G). A graph G is an extreme geodesic graph if g(G) = ex(G), that is, if every vertex lies on a u–v geodesic for some pair u, v of extreme vertices. It is shown that every pair a, b of integers with 0 a b is realizable as the extreme order and geodetic number, respectively, of some graph. For positive integers r, d, and k 2, it is shown that there exists an extreme geodesic graph G of radius r, diameter d, and geodetic number k. Also, for integers n, d, and k with 2 d > n, 2 k > n, and n – d – k + 1 0, there exists a connected extreme geodesic graph G of order n, diameter d, and geodetic number k. We show that every graph of order n with geodetic number n – 1 is an extreme geodesic graph. On the other hand, for every pair k, n of integers with 2 k n – 2, there exists a connected graph of order n with geodetic number k that is not an extreme geodesic graph.     

On pitfalls in computing the geodetic number of a graph

Given a simple connected graph G = (V, E) the geodetic closure of a subset S of V is the union of all sets of nodes lying on some geodesic (or shortest path) joining a pair of nodes . The geodetic number, denoted by g(G), is the smallest cardinality of a node set S ^* such that I[S ^*] = V. In “The geodetic number of a graph”, [Harary et al. in Math. Comput. Model. 17:89–95, 1993] propose an incorrect algorithm to find the geodetic number of a graph G. We provide counterexamples and show why the proposed approach must fail. We then develop a 0–1 integer programming model to find the geodetic number. Computational results are given.     

The Forcing Convexity Number of a Graph

For two vertices u and v of a connected graph G, the set I(u,v) consists of all those vertices lying on a u-v geodesic in G. For a set S of vertices of G, the union of all sets I(u,v) for u, v S is denoted by I(S). A set S is a convex set if I(S) = S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. A convex set S in G with |S| = con(G) is called a maximum convex set. A subset T of a maximum convex set S of a connected graph G is called a forcing subset for S if S is the unique maximum convex set containing T. The forcing convexity number f(S, con) of S is the minimum cardinality among the forcing subsets for S, and the forcing convexity number f(G, con) of G is the minimum forcing convexity number among all maximum convex sets of G. The forcing convexity numbers of several classes of graphs are presented, including complete bipartite graphs, trees, and cycles. For every graph G, f(G, con) con(G). It is shown that every pair a, b of integers with 0 a b and b is realizable as the forcing convexity number and convexity number, respectively, of some connected graph. The forcing convexity number of the Cartesian product of H × K ₂ for a nontrivial connected graph H is studied.     

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.     

Convexity in oriented graphs

For vertices u and v in an oriented graph D, the closed interval I[u,v] consists of u and v together with all vertices lying in a u−v geodesic or v−u geodesic in D. For S⊆V(D), I[S] is the union of all closed intervals I[u,v] with u,v∈S. A set S is convex if I[S]=S. The convexity number con(D) is the maximum cardinality of a proper convex set of V(D). The nontrivial connected oriented graphs of order n with convexity number n−1 are characterized. It is shown that there is no connected oriented graph of order at least 4 with convexity number 2 and that every pair k, n of integers with 1⩽k⩽n−1 and k≠2 is realizable as the convexity number and order, respectively, of some connected oriented graph. For a nontrivial connected graph G, the lower orientable convexity number con⁻(G) is the minimum convexity number among all orientations of G and the upper orientable convexity number con⁺(G) is the maximum such convexity number. It is shown that con⁺(G)=n−1 for every graph G of order n⩾2. The lower orientable convexity numbers of some well-known graphs are determined, with special attention given to outerplanar graphs.     

The Convexity Number of a Graph

 For two vertices u and v of a connected graph G, the set I[u,v] consists of all those vertices lying on a u−v shortest path in G, while for a set S of vertices of G, the set I[S] is the union of all sets I[u,v] for u,v∈S. A set S is convex if I[S]=S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. The clique number ω(G) is the maximum cardinality of a clique in G. If G is a connected graph of order n that is not complete, then n≥3 and 2≤ω(G)≤con(G)≤n−1. It is shown that for every triple l,k,n of integers with n≥3 and 2≤l≤k≤n−1, there exists a noncomplete connected graph G of order n with ω(G)=l and con(G)=k. Other results on convex numbers are also presented. Received: August 19, 1998 Final version received: May 17, 2000     

On geodetic sets formed by boundary vertices

Let G be a finite simple connected graph. A vertex v is a boundary vertex of G if there exists a vertex u such that no neighbor of v is further away from u than v. We obtain a number of properties involving different types of boundary vertices: peripheral, contour and eccentric vertices. Before showing that one of the main results in [G. Chartrand, D. Erwin, G.L. Johns, P. Zhang, Boundary vertices in graphs, Discrete Math. 263 (2003) 25-34] does not hold for one of the cases, we establish a realization theorem that not only corrects the mentioned wrong statement but also improves it.Given S⊆V(G), its geodetic closure I[S] is the set of all vertices lying on some shortest path joining two vertices of S. We prove that the boundary vertex set ∂(G) of any graph G is geodetic, that is, I[∂(G)]=V(G). A vertex v belongs to the contour Ct(G) of G if no neighbor of v has an eccentricity greater than v. We present some sufficient conditions to guarantee the geodeticity of either the contour Ct(G) or its geodetic closure I[Ct(G)].     

Geodesics and almost geodesic cycles in random regular graphs

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d_G(u, v) is at least d_C(u, v)?e(n). Let ω(n) be any function tending to infinity with n. We consider a random d‐regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n) = log_d?1log_d?1n+ ω(n) and |C| = 2log_d?1n+ O(ω(n)). Along the way, we obtain results on near‐geodesic paths. We also give the limiting distribution of the number of geodesics between two random vertices in this random graph. Copyright © 2010 John Wiley & Sons, Ltd. J Graph Theory 66:115‐136, 2011     

On the computation of the hull number of a graph

Let G be a graph. If u,v∈V(G), a u-vshortest path of G is a path linking u and v with minimum number of edges. The closed interval I[u,v] consists of all vertices lying in some u-v shortest path of G. For S⊆V(G), the set I[S] is the union of all sets I[u,v] for u,v∈S. We say that S is a convex set if I[S]=S. The convex hull of S, denoted I_h[S], is the smallest convex set containing S. A set S is a hull set of G if I_h[S]=V(G). The cardinality of a minimum hull set of G is the hull number of G, denoted by hn(G). In this work we prove that deciding whether hn(G)≤k is NP-complete.We also present polynomial-time algorithms for computing hn(G) when G is a unit interval graph, a cograph or a split graph.     

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.     

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.     

Isometric cycles,cutsets, and crowning of bridged graphs

A graph G is bridged if every cycle C of length at least 4 has vertices x,y such that d_G(x,y) < d_C(x,y). A cycle C is isometric if d_G(x,y) = d_C(x,y) for all x,y ∈ V(C). We show that every graph contractible to a graph with girth g has an isometric cycle of length at least g. We use this to show that every minimal cutset S in a bridged graph G induces a connected subgraph. We introduce a “crowning” construction to enlarge bridged graphs. We use this to construct examples showing that for every connected simple graph H with girth at least 6 (including trees), there exists a bridged graph G such that G has a unique minimum cutset S and that G[S] = H. This provides counterexamples to Hahn's conjecture that d_G(u,v) ≤ 2 when u and v lie in a minimum cutset in a bridged graph G. We also study the convexity of cutsets in bridged graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 161–170, 2003     

The Sigma Chromatic Number of a Graph

For a nontrivial connected graph G, let ${c: V(G)\to {{\mathbb N}}}For a nontrivial connected graph G, let c: V(G)? \mathbb N{c: V(G)\to {{\mathbb N}}} be a vertex coloring of G, where adjacent vertices may be colored the same. For a vertex v of G, let N(v) denote the set of vertices adjacent to v. The color sum σ(v) of v is the sum of the colors of the vertices in N(v). If σ(u) ≠ σ(v) for every two adjacent vertices u and v of G, then c is called a sigma coloring of G. The minimum number of colors required in a sigma coloring of a graph G is called its sigma chromatic number σ(G). The sigma chromatic number of a graph G never exceeds its chromatic number χ(G) and for every pair a, b of positive integers with a ≤ b, there exists a connected graph G with σ(G) = a and χ(G) = b. There is a connected graph G of order n with σ(G) = k for every pair k, n of positive integers with k ≤n if and only if k ≠ n − 1. Several other results concerning sigma chromatic numbers are presented.     

An algebraic characterization of geodetic graphs

We say that a binary operation * is associated with a (finite undirected) graph G (without loops and multiple edges) if * is defined on V(G) and uv E(G) if and only if u v, u * v = v and v * u = u for any u, v V(G). In the paper it is proved that a connected graph G is geodetic if and only if there exists a binary operation associated with G which fulfils a certain set of four axioms. (This characterization is obtained as an immediate consequence of a stronger result proved in the paper).