Exact algorithms for weak Roman domination

Abstract

We consider the Weak Roman Domination problem. Given an undirected graph G=(V,E), the aim is to find a weak Roman domination function (wrd-function for short) of minimum cost, i.e. a function f:V→{0,1,2} such that every vertex v∈V is defended (i.e. there exists a neighbor u of v, possibly u=v, such that f(u)⩾1) and for every vertex v∈V with f(v)=0 there exists a neighbor u of v such that f(u)⩾1 and the function fu→v defined by fu→v(v)=1, fu→v(u)=f(u)−1 and fu→v(x)=f(x) otherwise does not contain any undefended vertex. The cost of a wrd-function f is defined by cost(f)=∑v∈Vf(v). The trivial enumeration algorithm runs in time O∗(3n) and polynomial space and is the best one known for the problem so far. We are breaking the trivial enumeration barrier by providing two faster algorithms: we first prove that the problem can be solved in O∗(2n) time needing exponential space, and then describe an O∗(2.2279n) algorithm using polynomial space. Our results rely on structural properties of a wrd-function, as well as on the best polynomial space algorithm for the Red-Blue Dominating Set problem. Moreover we show that the problem can be solved in linear-time on interval graphs.