15,887 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

    Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs

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    A total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set. We introduce and study graphs that admit non-negative real weights associated to their vertices such that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We show that these graphs, which we call total domishold graphs, form a non-hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs, and are closely related to threshold Boolean functions and threshold hypergraphs. We present a polynomial time recognition algorithm of total domishold graphs, and characterize graphs in which the above property holds in a hereditary sense. Our characterization is obtained by studying a new family of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of independent interest.Comment: 19 pages, 1 figur

    Degree Sequence Index Strategy

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    We introduce a procedure, called the Degree Sequence Index Strategy (DSI), by which to bound graph invariants by certain indices in the ordered degree sequence. As an illustration of the DSI strategy, we show how it can be used to give new upper and lower bounds on the kk-independence and the kk-domination numbers. These include, among other things, a double generalization of the annihilation number, a recently introduced upper bound on the independence number. Next, we use the DSI strategy in conjunction with planarity, to generalize some results of Caro and Roddity about independence number in planar graphs. Lastly, for claw-free and K1,rK_{1,r}-free graphs, we use DSI to generalize some results of Faudree, Gould, Jacobson, Lesniak and Lindquester
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