7 research outputs found
Balanced Independent and Dominating Sets on Colored Interval Graphs
We study two new versions of independent and dominating set problems on
vertex-colored interval graphs, namely \emph{-Balanced Independent Set}
(-BIS) and \emph{-Balanced Dominating Set} (-BDS). Let be a
vertex-colored interval graph with a -coloring for some . A subset of vertices
is called \emph{-balanced} if contains vertices from each color
class. In the -BIS and -BDS problems, the objective is to compute an
independent set or a dominating set that is -balanced. We show that both
problems are \NP-complete even on proper interval graphs. For the BIS problem
on interval graphs, we design two \FPT\ algorithms, one parameterized by
and the other by the vertex cover number of . Moreover, we present a
2-approximation algorithm for a slight variation of BIS on proper interval
graphs
Listing Maximal Independent Sets with Minimal Space and Bounded Delay
International audienceAn independent set is a set of nodes in a graph such that no two of them are adjacent. It is maximal if there is no node outside the independent set that may join it. Listing maximal independent sets in graphs can be applied, for example, to sample nodes belonging to different communities or clusters in network analysis and document clustering. The problem has a rich history as it is related to maximal cliques, dominance sets, vertex covers and 3-colorings in graphs. We are interested in reducing the delay, which is the worst-case time between any two consecutively output solutions, and the memory footprint, which is the additional working space behind the read-only input graph
Counting the number of independent sets in chordal graphs
We study some counting and enumeration problems for chordal graphs, especially concerning independent sets. We first provide the following efficient algorithms for a chordal graph: (1) a linear-time algorithm for counting the number of independent sets; (2) a linear-time algorithm for counting the number of maximum independent sets; (3) a polynomial-time algorithm for counting the number of independent sets of a fixed size. With similar ideas, we show that enumeration (namely, listing) of the independent sets, the maximum independent sets, and the independent sets of a fixed size in a chordal graph can be done in constant amortized time per output. On the other hand, we prove that the following problems for a chordal graph are #P-complete: (1) counting the number of maximal independent sets; (2) counting the number of minimum maximal independent sets. With similar ideas, we also show that finding a minimum weighted maximal independent set in a chordal graph is NP-hard, and even hard to approximate