6,802 research outputs found
Improved Algorithms for Recognizing Perfect Graphs and Finding Shortest Odd and Even Holes
Various classes of induced subgraphs are involved in the deepest results of
graph theory and graph algorithms. A prominent example concerns the {\em
perfection} of that the chromatic number of each induced subgraph of
equals the clique number of . The seminal Strong Perfect Graph Theorem
confirms that the perfection of can be determined by detecting odd holes in
and its complement. Chudnovsky et al. show in 2005 an algorithm
for recognizing perfect graphs, which can be implemented to run in
time for the exponent of square-matrix
multiplication. We show the following improved algorithms.
1. The tractability of detecting odd holes was open for decades until the
major breakthrough of Chudnovsky et al. in 2020. Their algorithm is
later implemented by Lai et al. to run in time, leading to the best
formerly known algorithm for recognizing perfect graphs. Our first result is an
algorithm for detecting odd holes, implying an algorithm for
recognizing perfect graphs.
2. Chudnovsky et al. extend in 2021 the algorithms for detecting odd
holes (2020) and recognizing perfect graphs (2005) into the first polynomial
algorithm for obtaining a shortest odd hole, which runs in time. We
reduce the time for finding a shortest odd hole to .
3. Conforti et al. show in 1997 the first polynomial algorithm for detecting
even holes, running in about time. It then takes a line of
intensive efforts in the literature to bring down the complexity to
, , , and finally . On the other hand,
the tractability of finding a shortest even hole has been open for 16 years
until the very recent algorithm of Cheong and Lu in 2022. We
improve the time of finding a shortest even hole to .Comment: 29 pages, 5 figure
On Polar, Trivially Perfect Graphs
During the last decades, different types of decompositions have been processed in the field of graph theory. In various problems, for example in the construction of recognition algorithms, frequently appears the so-called weakly decomposition of graphs.Polar graphs are a natural extension of some classes of graphs like bipartite graphs, split graphs and complements of bipartite graphs. Recognizing a polar graph is known to be NP-complete. For this class of graphs, polynomial algorithms for the maximum stable set problem are unknown and algorithms for the dominating set problem are also NP-complete.In this paper we characterize the polar graphs using the weakly decomposition, give a polynomial time algorithm for recognizing graphs that are both trivially perfect and polar, and directly calculate the domination number. For the stability number and clique number, we give polynomial time algorithms.
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
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