17,388 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 n2n \ge 2 and maximum degree Δk\Delta \ge k, then Fk(G)(Δk+1)nΔk+1+min{δ,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)ΔnΔ+1Z(G)=F_1(G) \le \frac{\Delta n}{\Delta+1}. Moreover, when Δ2\Delta \ge 2 and the graph is kk-connected, we prove that Fk(G)(Δ2)n+2Δ+k2F_k(G) \leq \frac{(\Delta-2)n+2}{\Delta+k-2}, which is an improvement when k2k\leq 2, and specializes to, for the zero forcing number case, Z(G)=F1(G)(Δ2)n+2Δ1Z(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

    Protecting a Graph with Mobile Guards

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    Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

    Exponential Domination in Subcubic Graphs

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    As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduce exponential domination, where vertices are considered to have some dominating power that decreases exponentially with the distance, and the dominated vertices have to accumulate a sufficient amount of this power emanating from the dominating vertices. More precisely, if SS is a set of vertices of a graph GG, then SS is an exponential dominating set of GG if vS(12)dist(G,S)(u,v)11\sum\limits_{v\in S}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}\geq 1 for every vertex uu in V(G)SV(G)\setminus S, where dist(G,S)(u,v){\rm dist}_{(G,S)}(u,v) is the distance between uV(G)Su\in V(G)\setminus S and vSv\in S in the graph G(S{v})G-(S\setminus \{ v\}). The exponential domination number γe(G)\gamma_e(G) of GG is the minimum order of an exponential dominating set of GG. In the present paper we study exponential domination in subcubic graphs. Our results are as follows: If GG is a connected subcubic graph of order n(G)n(G), then n(G)6log2(n(G)+2)+4γe(G)13(n(G)+2).\frac{n(G)}{6\log_2(n(G)+2)+4}\leq \gamma_e(G)\leq \frac{1}{3}(n(G)+2). For every ϵ>0\epsilon>0, there is some gg such that γe(G)ϵn(G)\gamma_e(G)\leq \epsilon n(G) for every cubic graph GG of girth at least gg. For every 0<α<23ln(2)0<\alpha<\frac{2}{3\ln(2)}, there are infinitely many cubic graphs GG with γe(G)3n(G)ln(n(G))α\gamma_e(G)\leq \frac{3n(G)}{\ln(n(G))^{\alpha}}. If TT is a subcubic tree, then γe(T)16(n(T)+2).\gamma_e(T)\geq \frac{1}{6}(n(T)+2). For a given subcubic tree, γe(T)\gamma_e(T) can be determined in polynomial time. The minimum exponential dominating set problem is APX-hard for subcubic graphs

    Hybridizing Non-dominated Sorting Algorithms: Divide-and-Conquer Meets Best Order Sort

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    Many production-grade algorithms benefit from combining an asymptotically efficient algorithm for solving big problem instances, by splitting them into smaller ones, and an asymptotically inefficient algorithm with a very small implementation constant for solving small subproblems. A well-known example is stable sorting, where mergesort is often combined with insertion sort to achieve a constant but noticeable speed-up. We apply this idea to non-dominated sorting. Namely, we combine the divide-and-conquer algorithm, which has the currently best known asymptotic runtime of O(N(logN)M1)O(N (\log N)^{M - 1}), with the Best Order Sort algorithm, which has the runtime of O(N2M)O(N^2 M) but demonstrates the best practical performance out of quadratic algorithms. Empirical evaluation shows that the hybrid's running time is typically not worse than of both original algorithms, while for large numbers of points it outperforms them by at least 20%. For smaller numbers of objectives, the speedup can be as large as four times.Comment: A two-page abstract of this paper will appear in the proceedings companion of the 2017 Genetic and Evolutionary Computation Conference (GECCO 2017
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