562 research outputs found

    Upper bounds on the k-forcing number of a graph

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    Given a simple undirected graph GG and a positive integer kk, the kk-forcing number of GG, denoted Fk(G)F_k(G), is the minimum number of vertices that need to be initially colored so that all vertices eventually become colored during the discrete dynamical process described by the following rule. Starting from an initial set of colored vertices and stopping when all vertices are colored: if a colored vertex has at most kk non-colored neighbors, then each of its non-colored neighbors becomes colored. When k=1k=1, this is equivalent to the zero forcing number, usually denoted with Z(G)Z(G), a recently introduced invariant that gives an upper bound on the maximum nullity of a graph. In this paper, we give several upper bounds on the kk-forcing number. Notable among these, we show that if GG is a graph with order nβ‰₯2n \ge 2 and maximum degree Ξ”β‰₯k\Delta \ge k, then Fk(G)≀(Ξ”βˆ’k+1)nΞ”βˆ’k+1+min⁑{Ξ΄,k}F_k(G) \le \frac{(\Delta-k+1)n}{\Delta - k + 1 +\min{\{\delta,k\}}}. This simplifies to, for the zero forcing number case of k=1k=1, Z(G)=F1(G)≀ΔnΞ”+1Z(G)=F_1(G) \le \frac{\Delta n}{\Delta+1}. Moreover, when Ξ”β‰₯2\Delta \ge 2 and the graph is kk-connected, we prove that Fk(G)≀(Ξ”βˆ’2)n+2Ξ”+kβˆ’2F_k(G) \leq \frac{(\Delta-2)n+2}{\Delta+k-2}, which is an improvement when k≀2k\leq 2, and specializes to, for the zero forcing number case, Z(G)=F1(G)≀(Ξ”βˆ’2)n+2Ξ”βˆ’1Z(G)= F_1(G) \le \frac{(\Delta -2)n+2}{\Delta -1}. These results resolve a problem posed by Meyer about regular bipartite circulant graphs. Finally, we present a relationship between the kk-forcing number and the connected kk-domination number. As a corollary, we find that the sum of the zero forcing number and connected domination number is at most the order for connected graphs.Comment: 15 pages, 0 figure

    Disjunctive Total Domination in Graphs

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    Let GG be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, Ξ³t(G)\gamma_t(G). A set SS of vertices in GG is a disjunctive total dominating set of GG if every vertex is adjacent to a vertex of SS or has at least two vertices in SS at distance2 from it. The disjunctive total domination number, Ξ³td(G)\gamma^d_t(G), is the minimum cardinality of such a set. We observe that Ξ³td(G)≀γt(G)\gamma^d_t(G) \le \gamma_t(G). We prove that if GG is a connected graph of ordernβ‰₯8n \ge 8, then Ξ³td(G)≀2(nβˆ’1)/3\gamma^d_t(G) \le 2(n-1)/3 and we characterize the extremal graphs. It is known that if GG is a connected claw-free graph of ordernn, then Ξ³t(G)≀2n/3\gamma_t(G) \le 2n/3 and this upper bound is tight for arbitrarily largenn. We show this upper bound can be improved significantly for the disjunctive total domination number. We show that if GG is a connected claw-free graph of ordern>10n > 10, then Ξ³td(G)≀4n/7\gamma^d_t(G) \le 4n/7 and we characterize the graphs achieving equality in this bound.Comment: 23 page

    Linear-Delay Enumeration for Minimal Steiner Problems

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    Kimelfeld and Sagiv [Kimelfeld and Sagiv, PODS 2006], [Kimelfeld and Sagiv, Inf. Syst. 2008] pointed out the problem of enumerating KK-fragments is of great importance in a keyword search on data graphs. In a graph-theoretic term, the problem corresponds to enumerating minimal Steiner trees in (directed) graphs. In this paper, we propose a linear-delay and polynomial-space algorithm for enumerating all minimal Steiner trees, improving on a previous result in [Kimelfeld and Sagiv, Inf. Syst. 2008]. Our enumeration algorithm can be extended to other Steiner problems, such as minimal Steiner forests, minimal terminal Steiner trees, and minimal directed Steiner trees. As another variant of the minimal Steiner tree enumeration problem, we study the problem of enumerating minimal induced Steiner subgraphs. We propose a polynomial-delay and exponential-space enumeration algorithm of minimal induced Steiner subgraphs on claw-free graphs. Contrary to these tractable results, we show that the problem of enumerating minimal group Steiner trees is at least as hard as the minimal transversal enumeration problem on hypergraphs

    Extremal Infinite Graph Theory

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    We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure
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