124 research outputs found
Maximum Weight Independent Sets in Odd-Hole-Free Graphs Without Dart or Without Bull
The Maximum Weight Independent Set (MWIS) Problem on graphs with vertex
weights asks for a set of pairwise nonadjacent vertices of maximum total
weight. Being one of the most investigated and most important problems on
graphs, it is well known to be NP-complete and hard to approximate. The
complexity of MWIS is open for hole-free graphs (i.e., graphs without induced
subgraphs isomorphic to a chordless cycle of length at least five). By applying
clique separator decomposition as well as modular decomposition, we obtain
polynomial time solutions of MWIS for odd-hole- and dart-free graphs as well as
for odd-hole- and bull-free graphs (dart and bull have five vertices, say
, and dart has edges , while bull has edges
). If the graphs are hole-free instead of odd-hole-free then
stronger structural results and better time bounds are obtained
The Dilworth Number of Auto-Chordal-Bipartite Graphs
The mirror (or bipartite complement) mir(B) of a bipartite graph B=(X,Y,E)
has the same color classes X and Y as B, and two vertices x in X and y in Y are
adjacent in mir(B) if and only if xy is not in E. A bipartite graph is chordal
bipartite if none of its induced subgraphs is a chordless cycle with at least
six vertices. In this paper, we deal with chordal bipartite graphs whose mirror
is chordal bipartite as well; we call these graphs auto-chordal bipartite
graphs (ACB graphs for short). We describe the relationship to some known graph
classes such as interval and strongly chordal graphs and we present several
characterizations of ACB graphs. We show that ACB graphs have unbounded
Dilworth number, and we characterize ACB graphs with Dilworth number k
Independent sets of maximum weight in apple-free graphs
We present the first polynomial-time algorithm to solve the maximum weight independent set problem for apple-free graphs, which is a common generalization of several important classes where the problem can be solved efficiently, such as claw-free graphs, chordal graphs, and cographs. Our solution is based on a combination of two algorithmic techniques (modular decomposition and decomposition by clique separators) and a deep combinatorial analysis of the structure of apple-free graphs. Our algorithm is robust in the sense that it does not require the input graph G to be apple-free; the algorithm either finds an independent set of maximum weight in G or reports that G is not apple-free
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