DOUBLE EDGE-VERTEX DOMINATION IN GRAPHS: COMPLEXITY AND ALGORITHMS

A vertex u of a graph G = (V, E) is edge-vertex dominated by an edge e is an element of E if u is incident with e, or u is adjacent to a vertex incident with e. A subset D subset of E is a double edge-vertex dominating set of G if every vertex of G is edge-vertex dominated by at least two edges of D. The double edge-vertex domination number gamma(dev)(G) of G is the minimum cardinality of a double edge-vertex dominating set of G. In this paper, we first show that the decision problem corresponding to the problem of computing gamma(dev)(G) is NP-complete for bipartite and perfect elimination bipartite graphs. Then we present a linear algorithm for computing the double edge-vertex domination number for chain graphs and trees. Finally, we propose a Delta(G) approximation algorithm for the problem of findding a minimum double edge-vertex dominating set of a graph G, where Delta(G) is the maximum degree of G. Moreover, we prove that this problem cannot be approximated within (1-epsilon) ln vertical bar V vertical bar for any epsilon > 0 unless NP subset of DTIME (vertical bar V vertical bar(O(log log) (vertical bar V vertical bar)).