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A vertex subset S of a graph G = (V , E) is a [ 1 , 2 ] -dominating set if each vertex of V ∖ S is adjacent to either one or two vertices in S. The minimum cardinality of a [ 1 , 2 ] -dominating set of G , denoted by γ [ 1 , 2 ] (G) , is called the [ 1 , 2 ] -domination number of G. In this paper, the [ 1 , 2 ] -domination and the [ 1 , 2 ] -total domination numbers of the generalized Petersen graphs P (n , 2) are determined. [ABSTRACT FROM AUTHOR]

The triangle partition problem is a generalization of the well-known graph matching problem consisting of finding the maximum number of independent edges in a given graph, i.e., edges with no common node. Triangle partition instead aims to find the maximum number of disjoint triangles. The triangle partition problem is known to be NP-complete. Thus, in this paper, the focus is on the local maximization variant, called maximal triangle partition (MTP). Thus, paper presents a new self-stabilizing algorithm for MTP that converges in O( m) moves under the unfair distributed daemon. [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]