468 research outputs found
A linear algorithm for finding a minimum dominating set in a cactus
AbstractA dominating set in a graph G = (V,E) is a set of vertices D such that every vertex in V−D removal results in a disconnected graph. A block in a graph G is a maximal connected subgraph of G having no cutvertices. A cactus is a graph in which each block is either an edge or a cycle. In this paper we present a linear time algorithm for finding a minimum order dominating set in a cactus
Combinatorial Problems on -graphs
Bir\'{o}, Hujter, and Tuza introduced the concept of -graphs (1992),
intersection graphs of connected subgraphs of a subdivision of a graph .
They naturally generalize many important classes of graphs, e.g., interval
graphs and circular-arc graphs. We continue the study of these graph classes by
considering coloring, clique, and isomorphism problems on -graphs.
We show that for any fixed containing a certain 3-node, 6-edge multigraph
as a minor that the clique problem is APX-hard on -graphs and the
isomorphism problem is isomorphism-complete. We also provide positive results
on -graphs. Namely, when is a cactus the clique problem can be solved in
polynomial time. Also, when a graph has a Helly -representation, the
clique problem can be solved in polynomial time. Finally, we observe that one
can use treewidth techniques to show that both the -clique and list
-coloring problems are FPT on -graphs. These FPT results apply more
generally to treewidth-bounded graph classes where treewidth is bounded by a
function of the clique number
- …