20,659 research outputs found

    Efficient Enumeration of Dominating Sets for Sparse Graphs

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    A dominating set D of a graph G is a set of vertices such that any vertex in G is in D or its neighbor is in D. Enumeration of minimal dominating sets in a graph is one of central problems in enumeration study since enumeration of minimal dominating sets corresponds to enumeration of minimal hypergraph transversal. However, enumeration of dominating sets including non-minimal ones has not been received much attention. In this paper, we address enumeration problems for dominating sets from sparse graphs which are degenerate graphs and graphs with large girth, and we propose two algorithms for solving the problems. The first algorithm enumerates all the dominating sets for a k-degenerate graph in O(k) time per solution using O(n + m) space, where n and m are respectively the number of vertices and edges in an input graph. That is, the algorithm is optimal for graphs with constant degeneracy such as trees, planar graphs, H-minor free graphs with some fixed H. The second algorithm enumerates all the dominating sets in constant time per solution for input graphs with girth at least nine

    Distributed Connectivity Decomposition

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    We present time-efficient distributed algorithms for decomposing graphs with large edge or vertex connectivity into multiple spanning or dominating trees, respectively. As their primary applications, these decompositions allow us to achieve information flow with size close to the connectivity by parallelizing it along the trees. More specifically, our distributed decomposition algorithms are as follows: (I) A decomposition of each undirected graph with vertex-connectivity kk into (fractionally) vertex-disjoint weighted dominating trees with total weight Ω(klogn)\Omega(\frac{k}{\log n}), in O~(D+n)\widetilde{O}(D+\sqrt{n}) rounds. (II) A decomposition of each undirected graph with edge-connectivity λ\lambda into (fractionally) edge-disjoint weighted spanning trees with total weight λ12(1ε)\lceil\frac{\lambda-1}{2}\rceil(1-\varepsilon), in O~(D+nλ)\widetilde{O}(D+\sqrt{n\lambda}) rounds. We also show round complexity lower bounds of Ω~(D+nk)\tilde{\Omega}(D+\sqrt{\frac{n}{k}}) and Ω~(D+nλ)\tilde{\Omega}(D+\sqrt{\frac{n}{\lambda}}) for the above two decompositions, using techniques of [Das Sarma et al., STOC'11]. Moreover, our vertex-connectivity decomposition extends to centralized algorithms and improves the time complexity of [Censor-Hillel et al., SODA'14] from O(n3)O(n^3) to near-optimal O~(m)\tilde{O}(m). As corollaries, we also get distributed oblivious routing broadcast with O(1)O(1)-competitive edge-congestion and O(logn)O(\log n)-competitive vertex-congestion. Furthermore, the vertex connectivity decomposition leads to near-time-optimal O(logn)O(\log n)-approximation of vertex connectivity: centralized O~(m)\widetilde{O}(m) and distributed O~(D+n)\tilde{O}(D+\sqrt{n}). The former moves toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an O(m)O(m) centralized exact algorithm while the latter is the first distributed vertex connectivity approximation

    Dominating 2-broadcast in graphs: complexity, bounds and extremal graphs

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    Limited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded. As a natural extension of domination, we consider dominating 2-broadcasts along with the associated parameter, the dominating 2-broadcast number. We prove that computing the dominating 2-broadcast number is a NP-complete problem, but can be achieved in linear time for trees. We also give an upper bound for this parameter, that is tight for graphs as large as desired.Peer ReviewedPostprint (author's final draft

    On the algorithmic complexity of twelve covering and independence parameters of graphs

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    The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs

    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
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