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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]

We present a silent self-stabilizing distributed algorithm computing a maximal |$\ p$| -star decomposition of the underlying communication network. Under the unfair distributed scheduler, the most general scheduler model, the algorithm converges in at most |$12\Delta m + \mathcal{O}(m+n)$| moves, where |$m$| is the number of edges, |$n$| is the number of nodes and |$\Delta $| is the maximum node degree. Regarding the time complexity, we obtain the following results: our algorithm outperforms the previously known best algorithm by a factor of |$\Delta $| with respect to the move complexity. While the round complexity for the previous algorithm was unknown, we show a |$5\big \lfloor \frac{n}{p+1} \big \rfloor +5$| bound for our algorithm. [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]