Steiner centers and Steiner medians of graphs

Let G be a connected graph and S a set of vertices of G. The Steiner distance of S is the smallest number of edges in a connected subgraph of G that contains S and is denoted by d_G(S) or d(S). The Steiner n-eccentricity e_n(v) and Steiner n-distance d_n(v) of a vertex v in G are defined as e_n(v)=max{d(S)| SV(G), |S|=n and vS} and d_n(v)=∑{d(S)| SV(G), |S|=n and vS}, respectively. The Steiner n-center C_n(G) of G is the subgraph induced by the vertices of minimum n-eccentricity. The Steiner n-median M_n(G) of G is the subgraph induced by those vertices with minimum Steiner n-distance. Let T be a tree. Oellermann and Tian [O.R. Oellermann, S. Tian, Steiner centers in graphs, J. Graph Theory 14 (1990) 585–597] showed that C_n(T) is contained in C_n+1(T) for all n2. Beineke et al. [L.W. Beineke, O.R. Oellermann, R.E. Pippert, On the Steiner median of a tree, Discrete Appl. Math. 68 (1996) 249–258] showed that M_n(T) is contained in M_n+1(T) for all n2. Then, Oellermann [O.R. Oellermann, On Steiner centers and Steiner medians of graphs, Networks 34 (1999) 258–263] asked whether these containment relationships hold for general graphs. In this note we show that for every n2 there is an infinite family of block graphs G for which C_n(G)C_n+1(G). We also show that for each n2 there is a distance–hereditary graph G such that M_n(G)M_n+1(G). Despite these negative examples, we prove that if G is a block graph then M_n(G) is contained in M_n+1(G) for all n2. Further, a linear time algorithm for finding the Steiner n-median of a block graph is presented and an efficient algorithm for finding the Steiner n-distances of all vertices in a block graph is described.     

Steiner centers in graphs

The Steiner distance of a set S of vertices in a connected graph G is the minimum size among all connected subgraphs of G containing S. For n ≥ 2, the n-eccentricity e_n(ν) of a vertex ν of a graph G is the maximum Steiner distance among all sets S of n vertices of G that contains ν. The n-diameter of G is the maximum n-eccentricity among the vertices of G while the n-radius of G is the minimum n-eccentricity. The n-center of G is the subgraph induced by those vertices of G having minimum n-eccentricity. It is shown that every graph is the n-center of some graph. Several results on the n-center of a tree are established. In particular, it is shown that the n-center of a tree is a tree and those trees that are n-centers of trees are characterized.     

Rooted induced trees in triangle‐free graphs

For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of Gthat is a tree. Further, for a vertex v∈V(G), let t(G, v) denote the maximum number of vertices in an induced subgraph of Gthat is a tree, with the extra condition that the tree must contain v. The minimum of t(G) (t(G, v), respectively) over all connected triangle‐free graphs G(and vertices v∈V(G)) on nvertices is denoted by t₃(n) (t(n)). Clearly, t(G, v)?t(G) for all v∈V(G). In this note, we solve the extremal problem of maximizing |G| for given t(G, v), given that Gis connected and triangle‐free. We show that and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\ge t_3^{\ast}(n) = \lceil {\frac{1}{2}}(1+{\sqrt{8n-7}})\rceilFor a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of Gthat is a tree. Further, for a vertex v∈V(G), let t(G, v) denote the maximum number of vertices in an induced subgraph of Gthat is a tree, with the extra condition that the tree must contain v. The minimum of t(G) (t(G, v), respectively) over all connected triangle‐free graphs G(and vertices v∈V(G)) on nvertices is denoted by t₃(n) (t(n)). Clearly, t(G, v)?t(G) for all v∈V(G). In this note, we solve the extremal problem of maximizing |G| for given t(G, v), given that Gis connected and triangle‐free. We show that and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\ge t_3^{\ast}(n) = \lceil {\frac{1}{2}}(1+{\sqrt{8n-7}})\rceil$, improving a recent result by Fox, Loh and Sudakov. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 206–209, 2010     

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.     

Induced subgraphs of prescribed size

A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let q(G) denote the maximum number of vertices in a trivial subgraph of G. Motivated by an open problem of Erd?s and McKay we show that every graph G on n vertices for which q(G)≤ C log n contains an induced subgraph with exactly y edges, for every y between 0 and n^{δ (C)}. Our methods enable us also to show that under much weaker assumption, i.e., q(G) ≤ n/14, G still must contain an induced subgraph with exactly y edges, for every y between 0 and . © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 239–251, 2003     

Upper total domination in claw‐free 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 (other than itself). The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Γ_t(G). We establish bounds on Γ_t(G) for claw‐free graphs G in terms of the number n of vertices and the minimum degree δ of G. We show that if if , and if δ ≥ 5. The extremal graphs are characterized. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 148–158, 2003     

On graphs with subgraphs having large independence numbers

Let G be a graph on n vertices in which every induced subgraph on vertices has an independent set of size at least . What is the largest so that every such G must contain an independent set of size at least q? This is one of the several related questions raised by Erd?s and Hajnal. We show that , investigate the more general problem obtained by changing the parameters s and t, and discuss the connection to a related Ramsey‐type problem. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 149–157, 2007     

The w-median of a connected strongly chordal graph

Suppose G = (V, E) is a graph in which every vertex x has a non-negative real number w(x) as its weight. The w-distance sum of a vertex y is D_{G, w}(y) = σ_x?v d(y, x)w(x). The w-median of G is the set of all vertices y with minimum w-distance sum D_G,w(y). This paper shows that the w-median of a connected strongly chordal graph G is a clique when w(x) is positive for all vertices x in G.     

Eulerian subgraphs in 3‐edge‐connected graphs and Hamiltonian line graphs

In this paper, we show that if G is a 3‐edge‐connected graph with and , then either G has an Eulerian subgraph H such that , or G can be contracted to the Petersen graph in such a way that the preimage of each vertex of the Petersen graph contains at least one vertex in S. If G is a 3‐edge‐connected planar graph, then for any , G has an Eulerian subgraph H such that . As an application, we obtain a new result on Hamiltonian line graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 308–319, 2003     

Explicit sparse almost‐universal graphs for ${\bf {{\cal G}(n, {k \over n})}}$

For any integer n, let be a probability distribution on the family of graphs on n vertices (where every such graph has nonzero probability associated with it). A graph Γ is ‐almost‐universal if Γ satisifies the following: If G is chosen according to the probability distribution , then G is isomorphic to a subgraph of Γ with probability 1 ‐ . For any p ∈ [0,1], let (n,p) denote the probability distribution on the family of graphs on n vertices, where two vertices u and v form an edge with probability p, and the events {u and v form an edge}; u,v ∈ V (G) are mutually independent. For k ≥ 4 and n sufficiently large we construct a ‐almost‐universal‐graph on n vertices and with O(n)polylog(n) edges, where q = ? ? for such k ≤ 6, and where q = ? ? for k ≥ 7. The number of edges is close to the lower bound of Ω( ) for the number of edges in a universal graph for the family of graphs with n vertices and maximum degree k. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Random walks which prefer unvisited edges: Exploring high girth even degree expanders in linear time

Let be a connected graph with vertices. A simple random walk on the vertex set of G is a process, which at each step moves from its current vertex position to a neighbouring vertex chosen uniformly at random. We consider a modified walk which, whenever possible, chooses an unvisited edge for the next transition; and makes a simple random walk otherwise. We call such a walk an edge‐process (or E‐process). The rule used to choose among unvisited edges at any step has no effect on our analysis. One possible method is to choose an unvisited edge uniformly at random, but we impose no such restriction. For the class of connected even degree graphs of constant maximum degree, we bound the vertex cover time of the E‐process in terms of the edge expansion rate of the graph G, as measured by eigenvalue gap of the transition matrix of a simple random walk on G. A vertex v is ?‐good, if any even degree subgraph containing all edges incident with v contains at least ? vertices. A graph G is ?‐good, if every vertex has the ?‐good property. Let G be an even degree ?‐good expander of bounded maximum degree. Any E‐process on G has vertex cover time This is to be compared with the lower bound on the cover time of any connected graph by a weighted random walk. Our result is independent of the rule used to select the order of the unvisited edges, which could, for example, be chosen on‐line by an adversary. As no walk based process can cover an n vertex graph in less than n – 1 steps, the cover time of the E‐process is of optimal order when . With high probability random r‐regular graphs, even, have . Thus the vertex cover time of the E‐process on such graphs is . © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 36–54, 2015     

The Forwarding Indices of Random Graphs

A routing R of a graph G is a set of n(n ? 1) elementary paths R(u, v) specified for all ordered pairs (u, v) of vertices of G. The vertex-forwarding index ξ(G) of G, is defined by Where ξ(G, R) is the maximum number of paths of the routing R passing through any vertex of G and the minimum is taken over all the routings of G. Let G_p denote the random graph on n vertices with edge probability p and let m = np. It is proved among other things that, under natural growth conditions on the function p = p(n), the ratio Tends to 1 in probability as n tends to infinity.     

Cutting a graph into two dissimilar halves

Given a graph G and a subset S of the vertex set of G, the discrepancy of S is defined as the difference between the actual and expected numbers of the edges in the subgraph induced on S. We show that for every graph with n vertices and e edges, n < e < n(n ? 1)/4, there is an n/2-element subset with the discrepancy of the order of magnitude of \documentclass{article}\pagestyle{empty}\begin{document}$\sqrt {ne}$\end{document} For graphs with fewer than n edges, we calculate the asymptotics for the maximum guaranteed discrepancy of an n/2-element subset. We also introduce a new notion called “bipartite discrepancy” and discuss related results and open problems.     

Coloring graphs from random lists of fixed size

Let G = G(n) be a graph on n vertices with maximum degree bounded by some absolute constant Δ. Assign to each vertex v of G a list L(v) of colors by choosing each list uniformly at random from all k‐subsets of a color set of size . Such a list assignment is called a random ‐list assignment. In this paper, we are interested in determining the asymptotic probability (as ) of the existence of a proper coloring ? of G, such that for every vertex v of G. We show, for all fixed k and growing n, that if , then the probability that G has such a proper coloring tends to 1 as . A similar result for complete graphs is also obtained: if and L is a random ‐list assignment for the complete graph K_n on n vertices, then the probability that K_n has a proper coloring with colors from the random lists tends to 1 as .Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 317‐327, 2014     

On 3-Steiner simplicial orderings

Let G be a connected graph and S a nonempty set of vertices of G. A Steiner tree for S is a connected subgraph of G containing S that has a minimum number of edges. The Steiner interval for S is the collection of all vertices in G that belong to some Steiner tree for S. Let k≥2 be an integer. A set X of vertices of G is k-Steiner convex if it contains the Steiner interval of every set of k vertices in X. A vertex x∈X is an extreme vertex of X if X?{x} is also k-Steiner convex. We call such vertices k-Steiner simplicial vertices. We characterize vertices that are 3-Steiner simplicial and give characterizations of two classes of graphs, namely the class of graphs for which every ordering produced by Lexicographic Breadth First Search is a 3-Steiner simplicial ordering and the class for which every ordering of every induced subgraph produced by Maximum Cardinality Search is a 3-Steiner simplicial ordering.     

Circular choosability via combinatorial Nullstellensatz

A p‐list assignment L of a graph G assigns to each vertex v of G a set of permissible colors. We say G is L‐(P, q)‐colorable if G has a (P, q)‐coloring h such that h(v) ? L(v) for each vertex v. The circular list chromatic number of a graph G is the infimum of those real numbers t for which the following holds: For any P, q, for any P‐list assignment L with , G is L‐(P, q)‐colorable. We prove that if G has an orientation D which has no odd directed cycles, and L is a P‐list assignment of G such that for each vertex v, , then G is L‐(P, q)‐colorable. This implies that if G is a bipartite graph, then , where is the maximum average degree of a subgraph of G. We further prove that if G is a connected bipartite graph which is not a tree, then . © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 190–204, 2008     

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.     

Coloring graphs of various maximum degree from random lists

Let G = G(n) be a graph on n vertices with maximum degree Δ =Δ (n). Assign to each vertex v of G a list L(v) of colors by choosing each list independently and uniformly at random from all k‐subsets of a color set of size . Such a list assignment is called a random ‐list assignment. In this paper, we are interested in determining the asymptotic probability (as n →∞) of the existence of a proper coloring φ of G, such that for every vertex v of G, a so‐called L‐coloring. We give various lower bounds on σ, in terms of n, k, and Δ, which ensures that with probability tending to 1 as n →∞ there is an L‐coloring of G. In particular, we show, for all fixed k and growing n, that if and , then the probability that G has an L‐coloring tends to 1 as . If and , then the same conclusion holds provided that . We also give related results for other bounds on Δ, when k is constant or a strictly increasing function of n.     

Note on independent sets in steiner systems

A partial Steiner (n, k, l)-system or briefly (n, k, l)-system is a pair (V, S), where V is an n-set and S is a collection of k-subsets of V, such that every l-subset of V is contained in at most one k-subset of S. A subset X ? V is called independent if [X]^k ∩ S = 0. The size of the largest independent set in S is denoted by α(S). Define The purpose of this note is to prove that for every k, l, k > l holds, where c, d are positive constants depending on k and l only.     

Families of pairs of graphs with a large number of common cards

The vertex‐deleted subgraph G?v, obtained from the graph G by deleting the vertex v and all edges incident to v, is called a card of G. The deck of G is the multiset of its unlabelled vertex‐deleted subgraphs. The number of common cards of G and H (or between G and H) is the cardinality of the multiset intersection of the decks of G and H. In this article, we present infinite families of pairs of graphs of order n ≥ 4 that have at least \begin{eqnarray*}2\lfloor\frac{1}{3}(n-1)\rfloor\end{eqnarray*} common cards; we conjecture that these, along with a small number of other families constructed from them, are the only pairs of graphs having this many common cards, for sufficiently large n. This leads us to propose a new stronger version of the Reconstruction Conjecture. In addition, we present an infinite family of pairs of graphs with the same degree sequence that have \begin{eqnarray*}\frac{2}{3}(n+5-2\sqrt{3n+6})\end{eqnarray*} common cards, for appropriate values of n, from which we can construct pairs having slightly fewer common cards for all other values of n≥10. We also present infinite families of pairs of forests and pairs of trees with \begin{eqnarray*}2\lfloor\frac{1}{3}(n-4)\rfloor\end{eqnarray*} and \begin{eqnarray*}2\lfloor\frac{1}{3}(n-5)\rfloor\end{eqnarray*} common cards, respectively. We then present new families that have the maximum number of common cards when one graph is connected and the other disconnected. Finally, we present a family with a large number of common cards, where one graph is a tree and the other unicyclic, and discuss how many cards are required to determine whether a graph is a tree. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 146–163, 2010