27,903 research outputs found

    Factor-Critical Property in 3-Dominating-Critical Graphs

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    A vertex subset SS of a graph GG is a dominating set if every vertex of GG either belongs to SS or is adjacent to a vertex of SS. The cardinality of a smallest dominating set is called the dominating number of GG and is denoted by γ(G)\gamma(G). A graph GG is said to be γ\gamma- vertex-critical if γ(Gv)<γ(G)\gamma(G-v)< \gamma(G), for every vertex vv in GG. Let GG be a 2-connected K1,5K_{1,5}-free 3-vertex-critical graph. For any vertex vV(G)v \in V(G), we show that GvG-v has a perfect matching (except two graphs), which is a conjecture posed by Ananchuen and Plummer.Comment: 8 page

    Zero forcing in iterated line digraphs

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    Zero forcing is a propagation process on a graph, or digraph, defined in linear algebra to provide a bound for the minimum rank problem. Independently, zero forcing was introduced in physics, computer science and network science, areas where line digraphs are frequently used as models. Zero forcing is also related to power domination, a propagation process that models the monitoring of electrical power networks. In this paper we study zero forcing in iterated line digraphs and provide a relationship between zero forcing and power domination in line digraphs. In particular, for regular iterated line digraphs we determine the minimum rank/maximum nullity, zero forcing number and power domination number, and provide constructions to attain them. We conclude that regular iterated line digraphs present optimal minimum rank/maximum nullity, zero forcing number and power domination number, and apply our results to determine those parameters on some families of digraphs often used in applications

    3-Factor-criticality in double domination edge critical graphs

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    A vertex subset SS of a graph GG is a double dominating set of GG if N[v]S2|N[v]\cap S|\geq 2 for each vertex vv of GG, where N[v]N[v] is the set of the vertex vv and vertices adjacent to vv. The double domination number of GG, denoted by γ×2(G)\gamma_{\times 2}(G), is the cardinality of a smallest double dominating set of GG. A graph GG is said to be double domination edge critical if γ×2(G+e)<γ×2(G)\gamma_{\times 2}(G+e)<\gamma_{\times 2}(G) for any edge eEe \notin E. A double domination edge critical graph GG with γ×2(G)=k\gamma_{\times 2}(G)=k is called kk-γ×2(G)\gamma_{\times 2}(G)-critical. A graph GG is rr-factor-critical if GSG-S has a perfect matching for each set SS of rr vertices in GG. In this paper we show that GG is 3-factor-critical if GG is a 3-connected claw-free 44-γ×2(G)\gamma_{\times 2}(G)-critical graph of odd order with minimum degree at least 4 except a family of graphs.Comment: 14 page

    An improved Belief Propagation algorithm finds many Bethe states in the random field Ising model on random graphs

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    We first present an empirical study of the Belief Propagation (BP) algorithm, when run on the random field Ising model defined on random regular graphs in the zero temperature limit. We introduce the notion of maximal solutions for the BP equations and we use them to fix a fraction of spins in their ground state configuration. At the phase transition point the fraction of unconstrained spins percolates and their number diverges with the system size. This in turn makes the associated optimization problem highly non trivial in the critical region. Using the bounds on the BP messages provided by the maximal solutions we design a new and very easy to implement BP scheme which is able to output a large number of stable fixed points. On one side this new algorithm is able to provide the minimum energy configuration with high probability in a competitive time. On the other side we found that the number of fixed points of the BP algorithm grows with the system size in the critical region. This unexpected feature poses new relevant questions on the physics of this class of models.Comment: 20 pages, 8 figure

    Edge-Stable Equimatchable Graphs

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    A graph GG is \emph{equimatchable} if every maximal matching of GG has the same cardinality. We are interested in equimatchable graphs such that the removal of any edge from the graph preserves the equimatchability. We call an equimatchable graph GG \emph{edge-stable} if GeG\setminus {e}, that is the graph obtained by the removal of edge ee from GG, is also equimatchable for any eE(G)e \in E(G). After noticing that edge-stable equimatchable graphs are either 2-connected factor-critical or bipartite, we characterize edge-stable equimatchable graphs. This characterization yields an O(min(n3.376,n1.5m))O(\min(n^{3.376}, n^{1.5}m)) time recognition algorithm. Lastly, we introduce and shortly discuss the related notions of edge-critical, vertex-stable and vertex-critical equimatchable graphs. In particular, we emphasize the links between our work and the well-studied notion of shedding vertices, and point out some open questions
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