Some classes of perfectly orderable graphs

In 1981, Chvátal defined the class of perfectly orderable graphs. This class of perfect graphs contains the comparability graphs and the triangulated graphs. In this paper, we introduce four classes of perfectly orderable graphs, including natural generalizations of the comparability and triangulated graphs. We provide recognition algorithms for these four classes. We also discuss how to solve the clique, clique cover, coloring, and stable set problems for these classes.     

Four classes of perfectly orderable graphs

A graph is called “perfectly orderable” if its vertices can be ordered in such a way that, for each induced subgraph F, a certain “greedy” coloring heuristic delivers an optimal coloring of F. No polynomial-time algorithm to recognize these graphs is known. We present four classes of perfectly orderable graphs: Welsh–Powell perfect graphs, Matula perfect graphs, graphs of Dilworth number at most three, and unions of two threshold graphs. Graphs in each of the first three classes are recognizable in a polynomial time. In every graph that belongs to one of the first two classes, we can find a largest clique and an optimal coloring in a linear time.     

On a conjecture of Hoàng and Tu concerning perfectly orderable graphs

Hoàng and Tu [On the perfect orderability of unions of two graphs, J. Graph Theory 33 (2000) 32-43] conjectured that a weakly triangulated graph which does not contain a chordless path with six vertices is perfectly orderable. We present a counter example to this conjecture.     

Which line-graphs are perfectly orderable?

We characterize (by forbidden induced subgraphs) those line-graphs that are perfectly orderable. Implicit in our presentation is a polynomial, time algorithm for recognizing these graphs.     

On perfectly two-edge connected graphs

This paper studies the graphs for which the 2-edge connected spanning subgraph polytope is completely described by the trivial inequalities and the so-called cut inequalities. These graphs are called perfectly 2-edge connected. The class of perfectly 2-edge connected graphs contains for instance the class of series-parallel graphs. We introduce a new class of perfectly 2-edge connected graphs. We discuss some structural properties of graphs which are (minimally with respect to some reduction operations) nonperfectly 2-edge connected. Using this we give sufficient conditions for a graph to be perfectly 2-edge connected.     

Uniquely orderable interval graphs

Interval graphs and interval orders are deeply linked. In fact, edges of an interval graphs represent the incomparability relation of an interval order, and in general, of different interval orders. The question about the conditions under which a given interval graph is associated to a unique interval order (up to duality) arises naturally. Fishburn provided a characterisation for uniquely orderable finite connected interval graphs. We show, by an entirely new proof, that the same characterisation holds also for infinite connected interval graphs. Using tools from reverse mathematics, we explain why the characterisation cannot be lifted from the finite to the infinite by compactness, as it often happens.     

On the complexity of recognizing directed path families

A Directed Path Family is a family of subsets of some finite ground set whose members can be realized as arc sets of simple directed paths in some directed graph. In this paper we show that recognizing whether a given family is a Directed Path family is an NP-Complete problem, even when all members in the family have at most two elements. If instead of a family of subsets, we are given a collection of words from some finite alphabet, then deciding whether there exists a directed graph G such that each word in the language is the set of arcs of some path in G, is a polynomial-time solvable problem.     

A characterization of edge-perfect graphs and the complexity of recognizing some combinatorial optimization games

We characterize edge-perfect graphs and prove that it is co-NP-complete to recognize them. In consequence, recognizing the defining matrices of totally balanced packing games is also co-NP-complete, in contrast with the polynomiality for the covering case. In addition, we solve the computational complexity of universally balanced (with respect to the resources constraints) packing games.     

Perfectly orderable graphs and almost all perfect graphs are kernelM-solvable

A graph is perfectly orderable if and only if it admits an acyclic orientation which does not contain an induced subgraph with verticesa, b, c, d and arcsab, bc, dc. Further a graph is called kernelM-solvable if for every direction of the edges (here pairs of symmetric, i.e. reversible, arcs are allowed) such that every directed triangle possesses at least two pairs of symmetric arcs, there exists a kernel, i.e. an independent setK of vertices such that every other vertex sends some arc towardsK. We prove that perfectly orderable graphs are kernelM-solvable. Using a deep result of Prömel and Steger we derive that almost all perfect graphs are kernelM-solvable.     

On the complexity of recognizing a class of generalized networks

The problem of determining whether a given linear programming problem can be converted to a generalized network flow problem having no unit-weight cycles is shown to be NP-hard. The same argument also shows that the problem of determining whether a gain matroid is bicircular is NP-hard.     

On recognizing and characterizing visibility graphs of simple polygons

In this paper we establish four necessary conditions for recognizing visibility graphs of simple polygons and conjecture that these conditions are sufficient. We present an 0(n²)-time algorithm for testing the first and second necessary conditions and leave it open whether the third and fourth necessary conditions can be tested in polynomial time. We also show that visibility graphs of simple polygons do not possess the characteristics of a few special classes of graphs Part of this work was done when the author visited the John Hopkins University and was Supported by NSF Grant DCR83-51468 and a grant from IBM.     

On the complexity of partitioning graphs into connected subgraphs

This paper is mainly concerned with the computational complexity of determining whether or not the vertices of a graph can be partitioned into equal sized subsets so that each subset induces a particular type of graph. Many of the NP-completeness results are for planar graphs. These are proved using a planar version of 3-dimensional matching.     

Reducing prime graphs and recognizing circle graphs

We prove a reduction theorem for prime (simple) graphs in Cunningham’s sense. Roughly speaking this theorem says that every prime (simple) graph of ordern>5 “contains” a smaller prime graph of ordern−1. As an application we give a polynomial algorithm for recognizing circle graphs.     

On the complexity of recognizing a set of vectors by a neuron

We consider the problems connected with the computational abilities of a neuron. The orderings of finite subsets of real vectors associated with neural computing are studied. We construct a lattice of such orderings and study some its properties. The interrelation between the orders on the sets and the neuron implementation of functions defined on these sets is derived. We prove the NP-hardness of “The Shortest Vector” problem and represent the relationship of the problem with neural computing.     

On the computational complexity of partial covers of Theta graphs

By use of elementary geometric arguments we prove the existence of a special integral solution of a certain system of linear equations. The existence of such a solution then yields the NP-hardness of the decision problem on the existence of locally injective homomorphisms to Theta graphs with three distinct odd path lengths.     

On the theory of right orderable groups

Translated from Matematicheskie Zametki, Vol. 54, No. 2, pp. 96–98, August, 1993.     

On the complexity of the independent set problem in triangle graphs

We consider the complexity of the maximum (maximum weight) independent set problem within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G−I, there is a vertex w∈I such that {u,v,w} is a triangle in G. We also introduce a new graph parameter (the upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal to the independence number. We prove that the problems under consideration are NP-complete, even for some restricted subclasses of triangle graphs, and provide several polynomially solvable cases for these problems within triangle graphs. Furthermore, we show that, for general triangle graphs, the maximum independent set problem and the upper independent neighborhood set problem cannot be polynomially approximated within any fixed constant factor greater than one unless P=NP.