10 research outputs found
Tropical Dominating Sets in Vertex-Coloured Graphs
Given a vertex-coloured graph, a dominating set is said to be tropical if
every colour of the graph appears at least once in the set. Here, we study
minimum tropical dominating sets from structural and algorithmic points of
view. First, we prove that the tropical dominating set problem is NP-complete
even when restricted to a simple path. Then, we establish upper bounds related
to various parameters of the graph such as minimum degree and number of edges.
We also give upper bounds for random graphs. Last, we give approximability and
inapproximability results for general and restricted classes of graphs, and
establish a FPT algorithm for interval graphs.Comment: 19 pages, 4 figure
Efficient Enumeration of Dominating Sets for Sparse Graphs
A dominating set D of a graph G is a set of vertices such that any vertex in G is in D or its neighbor is in D. Enumeration of minimal dominating sets in a graph is one of central problems in enumeration study since enumeration of minimal dominating sets corresponds to enumeration of minimal hypergraph transversal. However, enumeration of dominating sets including non-minimal ones has not been received much attention. In this paper, we address enumeration problems for dominating sets from sparse graphs which are degenerate graphs and graphs with large girth, and we propose two algorithms for solving the problems. The first algorithm enumerates all the dominating sets for a k-degenerate graph in O(k) time per solution using O(n + m) space, where n and m are respectively the number of vertices and edges in an input graph. That is, the algorithm is optimal for graphs with constant degeneracy such as trees, planar graphs, H-minor free graphs with some fixed H. The second algorithm enumerates all the dominating sets in constant time per solution for input graphs with girth at least nine
Tropical dominating sets in vertex-coloured graphs
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