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.

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.

scholarly journals Sobre Finura Própria de Grafos

2018 ◽  
Author(s):  
Moysés S. Sampaio Jr. ◽  
Fabiano S. Oliveira ◽  
Jayme L. Szwarcfiter
Keyword(s):  
Recognition Algorithm ◽  
Unit Interval ◽  
Interval Graphs ◽  
Graph Classes ◽  
Efficient Recognition

Both graph classes of k-thin and proper k-thin graphs have recently been introduced generalizing interval and unit interval graphs, respectively. The complexity of the recognition of k-thin and proper k-thin are open, even for fixed k 2. In this work, we introduce a subclass of the proper 2-thin graphs, called proper 2-thin of precedence. For this class, we present a characterization and an efficient recognition algorithm.

scholarly journals Classes of Graphs with $e$-Positive Chromatic Symmetric Function

10.37236/8211 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Angèle M. Foley ◽  
Chính T. Hoàng ◽  
Owen D. Merkel
Keyword(s):  
Symmetric Functions ◽  
Interval Graph ◽  
Unit Interval ◽  
Interval Graphs ◽  
Induced Subgraphs ◽  
Related Case ◽  
Graph Classes ◽  
Unit Interval Graph ◽  
Related Graph ◽  
Chromatic Symmetric Function

In the mid-1990s, Stanley and Stembridge conjectured that the chromatic symmetric functions of claw-free co-comparability (also called incomparability) graphs were $e$-positive. The quest for the proof of this conjecture has led to an examination of other, related graph classes. In 2013 Guay-Paquet proved that if unit interval graphs are $e$-positive, that implies claw-free incomparability graphs are as well. Inspired by this approach, we consider a related case and prove  that unit interval graphs whose complement is also a unit interval graph are $e$-positive.   We introduce the concept of strongly $e$-positive to denote a graph whose induced subgraphs are all $e$-positive, and conjecture that a graph is strongly $e$-positive if and only if it is (claw, net)-free.  

scholarly journals Classes of graphs with restricted interval models

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Andrzej Proskurowski ◽  
Jan Arne Telle
Keyword(s):  
Maximum Clique ◽  
Interval Graphs ◽  
Induced Subgraph ◽  
Graph Theoretic ◽  
Graph Classes ◽  
International Audience ◽  
Forbidden Induced Subgraph ◽  
Nested Hierarchy ◽  
Graph Families

International audience We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment.

scholarly journals $${\mathcal {U}}$$-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited

Algorithmica ◽  
2021 ◽  
Author(s):  
Jan Kratochvíl ◽  
Tomáš Masařík ◽  
Jana Novotná
Keyword(s):  
Unit Length ◽  
Algorithm Design ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Unit Interval ◽  
Interval Graphs ◽  
Induced Subgraph ◽  
Intersection Graphs ◽  
Proper Subclass ◽  
Maxcut Problem

AbstractInterval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs—a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a richer class of graphs. In particular, mixed unit interval graphs may contain a claw as an induced subgraph, as opposed to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyacı et al. (Inf Process Lett 121:29–33, 2017. 10.1016/j.ipl.2017.01.007). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs.

scholarly journals Multi-Armed Bandits on Partially Revealed Unit Interval Graphs

2020 ◽  
Vol 7 (3) ◽  
pp. 1453-1465 ◽  
Author(s):  
Xiao Xu ◽  
Sattar Vakili ◽  
Qing Zhao ◽  
Ananthram Swami
Keyword(s):  
Unit Interval ◽  
Interval Graphs
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