Exact algorithms for weak Roman domination

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 is an element of 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 is an element of V with f (v) = 0 there exists a neighbor u of v such that f (u) >= 1 and the function f(u -> v) defined by f(u -> v)(v) = 1, f(u -> v)(u) = f (u) - 1 and f(u -> v)(x) = f(x) otherwise does not contain any undefended vertex. The cost of a wrd-function f is defined by cost(f) = Sigma(v is an element of V)f(v). The trivial enumeration algorithm runs in time O*(3(n)) 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*(2(n)) time needing exponential space, and then describe an O*(2.2279(n)) 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. (C) 2017 Elsevier B.V. All rights reserved.