Extension Complexity of Stable Set Polytopes of Bipartite Graphs

A graph is a split graph if its vertex set can be partitioned into a clique and a stable set. A split graph is unbalanced if there exist two such partitions that are distinct. Cheng, Collins and Trenk (2016), discovered the following interesting counting fact: unlabeled, unbalanced split graphs on $n$ vertices can be placed into a bijection with all unlabeled split graphs on $n-1$ or fewer vertices. In this paper we translate these concepts and the theorem to different combinatorial settings: minimal set covers, bipartite graphs with a distinguished block and posets of height one.

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On the Steiner Tree 3/2-Approximation for Quasi-Bipartite Graphs

BRICS Report Series ◽

10.7146/brics.v6i39.20108 ◽

1999 ◽

Vol 6 (39) ◽

Cited By ~ 1

Author(s):

Romeo Rizzi

Keyword(s):

Approximation Algorithms ◽

Approximation Algorithm ◽

Local Search ◽

Key Words ◽

Steiner Tree ◽

Minimum Weight ◽

Bipartite Graphs ◽

Simple Graph ◽

Steiner Tree Problem ◽

Stable Set

Let G = (V,E) be an undirected simple graph and w : E -> R+ be a non-negative weighting of the edges of G. Assume V is partitioned as R union X. A Steiner tree is any tree T of G such that every node in R is incident with at least one edge of T. The metric Steiner tree problem asks for a Steiner tree of minimum weight, given that w is a metric. When X is a stable set of G, then (G,R,X) is called quasi-bipartite. In [1], Rajagopalan and Vazirani introduced the notion of quasi-bipartiteness and gave a ( 3/2 + epsilon) approximation algorithm for the metric Steiner tree problem, when (G,R,X) is quasi-bipartite. In this paper, we simplify and strengthen the result of Rajagopalan and Vazirani. We also show how classical bit scaling techniques can be adapted to the design of approximation algorithms.Key words: Steiner tree, local search, approximation algorithm, bit scaling.

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On Polar, Trivially Perfect Graphs

International Journal of Computers Communications & Control ◽

10.15837/ijccc.2010.5.2257 ◽

2010 ◽

Vol 5 (5) ◽

pp. 939

Author(s):

Mihai Talmaciu ◽

Elena Nechita

Keyword(s):

Polynomial Time ◽

Dominating Set ◽

Domination Number ◽

Natural Extension ◽

Polynomial Time Algorithm ◽

Clique Number ◽

Bipartite Graphs ◽

Stable Set ◽

Polynomial Algorithms ◽

Np Complete

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.

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