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Inspired by the split decomposition of graphs and rank-width, we introduce the notion of r-splits. We focus on the family of r-splits of a graph of order n, and we prove that it forms a hypergraph with several properties. We prove that such hypergraphs can be represented using only O(nr+1) of its hyperedges, despite its potentially exponential number of hyperedges. We also prove that there exist hypergraphs that need at least (nr) hyperedges to be represented, using a generalization of set orthogonality. [ABSTRACT FROM AUTHOR]

In computer networks area, the minimal dominating sets (MDS) and maximal independent sets (MIS) structures are very useful for creating virtual network overlays. Often, these set structures are used for designing efficient protocols in wireless sensor and ad-hoc networks. In this paper, we give a particular interest to one kind of these sets, called Independent Strong Dominating Set (ISD-set). In addition to its domination and independence properties, the ISD-set considers also node's degrees that make it very useful in practical applications where nodes with larger degrees play important role in the networks. For example, some network clustering protocols chose nodes with large degrees to be cluster-heads, which is exactly the result obtained by an ISD-set algorithm. Thence, we propose the first distributed self-stabilizing algorithm for computing an ISD-set of an arbitrary graph (called ISDS). Then, we prove that ISDS algorithm operates under the unfair distributed scheduler and converges after at most rounds requiring only space memory per node where Δ is the maximum node degree. The complexity of ISDS algorithm in rounds has the same order as the best known self-stabilizing algorithms for finding MDS and MIS. Moreover, performed simulations and comparisons with well-known self-stabilizing algorithms for MDS and MIS problems showed the efficiency of ISDS, especially for reducing the cardinality of dominating sets founded by the algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a new coloring problem of a graph called the dominated coloring. A dominated coloring of a graph $$G$$ is a proper vertex coloring of $$G$$ such that each color class is dominated by at least one vertex of $$G$$ . The minimum number of colors among all dominated colorings is called the dominated chromatic number, denoted by $$\chi _{dom}(G)$$ . In this paper, we establish the close relationship between the dominated chromatic number $$\chi _{dom}(G)$$ and the total domination number $$\gamma _t(G)$$ ; and the equivalence for triangle-free graphs. We study the complexity of the problem by proving its NP-completeness for arbitrary graphs having $$\chi _{dom}(G) \ge 4$$ and by giving a polynomial time algorithm for recognizing graphs having $$\chi _{dom}(G) \le 3$$ . We also give some bounds for planar and star-free graphs and exact values for split graphs. [ABSTRACT FROM AUTHOR]

Abstract: We consider the problem of matching unrooted unordered labeled trees, which refers to the task of evaluating the distance between trees. One of the most famous formalizations of this problem is the computation of the edit distance defined as the minimum-cost sequence of edit operations that transform one tree into another. Unfortunately, this problem has been proved to be NP-complete. In this paper, we propose a new algorithm to measure distance between unrooted unordered labeled trees. This algorithm uses a specific graph coloring to decompose the trees into small components (stars and bistars). Then, it determines a distance between two trees by computing the edit distance between their components. We prove that the proposed distance is a pseudo-metric and we analyze its time complexity. Our experimental evaluations on large synthetic and real world datasets confirm our analytical results and suggest that the distance we propose is accurate and its algorithm is scalable. [Copyright &y& Elsevier]