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Abstract: We introduce the vertex cover () problem, that is, the problem of finding a minimum weight set such that the graph has no , where is a path with vertices. The problem also has its application background. In this paper, we first show that the problem is NP-hard for any integer . Then we restrict our attention to the problem and give a 2-approximation algorithm using the primal–dual method. [Copyright &y& Elsevier]

Abstract: Let be a connected graph with vertices. Let be an integer. Suppose that a fire breaks out at a vertex of . A firefighter starts to protect vertices. At each time interval, the firefighter protects -vertices not yet on fire. At the end of each time interval, the fire spreads to all the unprotected vertices that have a neighbour on fire. Let sn denote the maximum number of vertices in that the firefighter can save when a fire breaks out at vertex . The -surviving rate of is defined to be , which is the average proportion of saved vertices. In this paper, we investigate the surviving rate of outerplanar graphs with vertices. The main results are as follows: (1) ; and (2) if , and if , which improves the result in [L.Cai, W.Wang, The surviving rate of a graph for the firefighter problem, SIAM J. Discrete Math. 23 (2009) 1814–1826]. [ABSTRACT FROM AUTHOR]

Abstract: Let be a graph without isolated vertices. For a positive integer , a set is a -distance paired-dominating set if each vertex in is within distance of a vertex in and the subgraph induced by contains a perfect matching. In this paper, we present two linear time algorithms to find a minimum cardinality -distance paired-dominating set in interval graphs and block graphs, which are two subclasses of chordal graphs. In addition, we present a characterization of trees with unique minimum -distance paired-dominating set. [Copyright &y& Elsevier]