BIPARTITE PERMUTATION GRAPHS ARE RECONSTRUCTIBLE

2012 ◽  
Vol 04 (03) ◽  
pp. 1250039
Author(s):  
MASASHI KIYOMI ◽  
TOSHIKI SAITOH ◽  
RYUHEI UEHARA
Keyword(s):  
Graph Theory ◽  
Planar Graphs ◽  
Unit Interval ◽  
Regular Graphs ◽  
Pendant Vertex ◽  
Permutation Graphs ◽  
Graph Classes ◽  
Graph Reconstruction ◽  
Bipartite Permutation Graphs ◽  
Disconnected Graphs

The graph reconstruction conjecture is a long-standing open problem in graph theory. The conjecture has been verified for all graphs with at most 11 vertices. Further, the conjecture has been verified for regular graphs, trees, disconnected graphs, unit interval graphs, separable graphs with no pendant vertex, outer-planar graphs, and unicyclic graphs. We extend the list of graph classes for which the conjecture holds. We give a proof that bipartite permutation graphs are reconstructible.

ON THE CLIQUE-WIDTH OF SOME PERFECT GRAPH CLASSES

2000 ◽  
Vol 11 (03) ◽  
pp. 423-443 ◽  
Author(s):  
MARTIN CHARLES GOLUMBIC ◽  
UDI ROTICS
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Unit Interval ◽  
Perfect Graphs ◽  
The Other ◽  
Perfect Graph ◽  
Permutation Graph ◽  
Permutation Graphs ◽  
Graph Classes ◽  
Unit Interval Graph

Graphs of clique–width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique–width of perfect graph classes. On one hand, we show that every distance–hereditary graph, has clique–width at most 3, and a 3–expression defining it can be obtained in linear time. On the other hand, we show that the classes of unit interval and permutation graphs are not of bounded clique–width. More precisely, we show that for every [Formula: see text] there is a unit interval graph In and a permutation graph Hn having n2 vertices, each of whose clique–width is at least n. These results allow us to see the border within the hierarchy of perfect graphs between classes whose clique–width is bounded and classes whose clique–width is unbounded. Finally we show that every n×n square grid, [Formula: see text], n ≥ 3, has clique–width exactly n+1.

scholarly journals On bipartite powers of bigraphs

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Yoshio Okamoto ◽  
Yota Otachi ◽  
Ryuhei Uehara
Keyword(s):  
Graph Theory ◽  
Permutation Graphs ◽  
International Audience ◽  
Bipartite Permutation Graphs ◽  
Np Complete

Graph Theory International audience The notion of graph powers is a well-studied topic in graph theory and its applications. In this paper, we investigate a bipartite analogue of graph powers, which we call bipartite powers of bigraphs. We show that the classes of bipartite permutation graphs and interval bigraphs are closed under taking bipartite power. We also show that the problem of recognizing bipartite powers is NP-complete in general.

Canonical Antichains of Unit Interval and Bipartite Permutation Graphs

Order ◽  
2010 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
Vadim V. Lozin ◽  
Colin Mayhill
Keyword(s):  
Unit Interval ◽  
Permutation Graphs ◽  
Bipartite Permutation Graphs

FAST EXPONENTIAL-TIME ALGORITHMS FOR THE FOREST COUNTING AND THE TUTTE POLYNOMIAL COMPUTATION IN GRAPH CLASSES

2009 ◽  
Vol 20 (01) ◽  
pp. 25-44 ◽  
Author(s):  
HEIDI GEBAUER ◽  
YOSHIO OKAMOTO
Keyword(s):  
Tutte Polynomial ◽  
Unit Interval ◽  
Interval Graphs ◽  
Chordal Graphs ◽  
Regular Graphs ◽  
Exponential Time ◽  
Graph Classes ◽  
Exponential Time Algorithms ◽  
Polynomial Factor

We prove # P -completeness for counting the number of forests in regular graphs and chordal graphs. We also present algorithms for this problem, running in O *(1.8494m) time for 3-regular graphs, and O *(1.9706m) time for unit interval graphs, where m is the number of edges in the graph and O *-notation ignores a polynomial factor. The algorithms can be generalized to the Tutte polynomial computation.

scholarly journals Subcritical Graph Classes Containing All Planar Graphs

2018 ◽  
Vol 27 (5) ◽  
pp. 763-773
Author(s):  
AGELOS GEORGAKOPOULOS ◽  
STEPHAN WAGNER
Keyword(s):  
Planar Graphs ◽  
Graph Classes

We construct minor-closed addable families of graphs that are subcritical and contain all planar graphs. This contradicts (one direction of) a well-known conjecture of Noy.

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