Class Of Perfect Graphs Research Articles

Path parity and perfection

Two nonadjacent vertices x and y in a graph G form an even pair if every induced path between them has an even number of edges. For a given pair { x, y} in a graph G, we denote by G xy the graph obtained from G by contracting x and y. In 1982, Fonlupt and Uhry proved that if G is perfect then so is G xy . In 1987, Meyniel used this fact to prove that no minimal imperfect graph contains an even pair. In the last eight years, even pairs have become an important tool for proving that certain classes of graphs are perfect and for designing optimization algorithms on special classes of perfect graphs. This paper surveys results of these types. It also discusses numerous related concepts including odd pairs.

Homomorphically full graphs

We study the class of graphs with the property that every homomorphic image is a subgraph. Our main theorem shows the equivalence of six different characterizations of these graphs. We establish some properties of this family of graphs, and relate them to other classes of perfect graphs.

Precoloring Extension III: Classes of Perfect Graphs

We continue the study of the following general problem on the vertex colourings of graphs. Suppose that some vertices of a graphGare assigned to some colours. Can this ‘precolouring’ be extended to a proper colouring ofGwith at mostkcolours (for some givenk)? Here we investigate the complexity status of precolouring extendibility on some classes of perfect graphs, giving good characterizations (necessary and sufficient conditions) that lead to algorithms with linear or polynomial running time. It is also shown how a larger subclass of perfect graphs can be derived from graphs containing no induced path on four vertices.

Pseudo‐Interval Graphs

AbstractWe study a class of perfect graphs which, because they generalize interval graphs, we call pseudo‐interval graphs. Like interval graphs, their vertices correspond to intervals of a linearly ordered set, but a modified definition of intersection is used in order to determine edges. The complements of pseudo‐interval graphs are comparability graphs but unlike interval graphs, pseudo‐interval graphs are only weakly triangulated. We characterize trees and complements of trees which are pseudo‐interval graphs. Finally we determine all minimal non‐pseudointerval graphs on eight or fewer vertices whose complements are comparability graphs and which are weakly triangulated. © 1995 John Wiley & Sons; Inc.

Constant-time parallel recognition of split graphs

Abstract We show that the class of perfect graphs known as split graphs can be efficiently recognized in a parallel process environment. Given an undirected graph G = (V, E) with n vertices we define a subgraph Ga = (AV, AE) and we prove some properties on Ga leading to a constant-time parallel split graph recognition algorithm.

Classes of graphs for which upper fractional domination equals independence, upper domination, and upper irredundance

This paper investigates cases where one graph parameter, upper fractional domination, is equal to three others: independence, upper domination and upper irredundance. We show that they are all equal for a large subclass, known as strongly perfect graphs, of the class of perfect graphs. They are also equal for odd cycles and upper bound graphs. However for simplicial graphs, upper irredundance might not equal the others, which are all equal. Also for many subclasses of perfect graphs other than the strongly perfect class, independence, upper domination and upper irredundance are not necessarily equal. We also show that if the graph join operation is used to combine two graphs which have some of the parameters equal, the resulting graph will have the same parameters equal.

Efficient algorithms for minimum weighted colouring of some classes of perfect graphs

We design an O( nm ) algorithm to find a minimum weighted colouring and a maximum weighted clique of a perfectly ordered graph. We also present two O( n 2 ) algorithms to find a minimum weighted colouring of a comparability graph and of a triangulated graph. Our colouring algorithms use an algorithm to find a stable set meeting all maximal (with respect to set inclusion) cliques of a perfectly ordered graph. We show that the problem of finding such stable set in an arbitrary graph is NP-hard. We shall also describe a polynomial algorithm to find a minimum weighted colouring of a clique separable graph.

Equistable graphs

AbstractAn equistable graph is a graph for which the incidence vectors of the maximal stable sets are the 0–1 solutions of a linear equation. A necessary condition and a sufficient condition for equistability are given. They are used to characterize the equistability of various classes of perfect graphs, outerplanar graphs, and pseudothreshold graphs. Some classes of equistable graphs are shown to be closed under graph substitution.

Polyominos and perfect graphs

Polyominos are useful for modelizing real-life problems. Many graphs can be associated with polyominos. Because some NP-problems become polynomial for the class of perfect graphs, to know whether or not these graphs are perfect is important from an algorithmic point of view. In this paper we shall investigate the perfection of several of these graphs.

Domination on Cocomparability Graphs

The authors determine the algorithmic complexity of domination and variants on cocomparability graphs, a class of perfect graphs containing both the interval and the permutation graphs. Minimum dominating, total dominating, connected dominating, and independent dominating sets can be constructed in polynomial time. On the other hand, DOMINATING CLIQUE and MINIMUM DOMINATING CLIQUE remain NP-complete on cocomparability graphs.

Perfect k‐line graphs and k‐total graphs

AbstractThe concept of the line graph can be generalized as follows. The k‐line graph Lk(G) of a graph G is defined as a graph whose vertices are the complete subgraphs on k vertices in G. Two distinct such complete subgraphs are adjacent in Lk(G) if and only if they have in G k − 1 vertices in common. The concept of the total graph can be generalized similarly. Then the Perfect Graph Conjecture will be proved for 3‐line graphs and 3‐total graphs. Moreover, perfect 3‐line graphs are not contained in any of the known classes of perfect graphs. © 1993 John Wiley & Sons, Inc.

A class of perfect graphs in genetics.

A straightforward graphical representation of a genealogy is regularized by chording cycles using only male-female links. The resulting graph is shown to be perfect.

S‐strongly perfect cartesian product of graphs

AbstractThe study of perfectness, via the strong perfect graph conjecture, has given rise to numerous investigations concerning the structure of many particular classes of perfect graphs. In “Perfect Product Graphs” (Discrete Mathematics, Vol. 20, 1977, pp. 177‐‐186), G. Ravindra and K. R. Parthasarathy tried, but unfortunately without success, to characterize the perfectness of the cartesian product of graphs (see also MR No. 58‐‐10567, 1979). In this paper we completely characterize the graphs that are both nontrivial cartesian products and s‐strongly perfect.

Two classes of perfect graphs

We present two classes of perfect graphs. The first class is defined through a construction used by Gallai in his investigation of comparability graphs; the second class contains those Berge graphs defined through a forbidden graph on five vertices.

Cosine: A new graph coloring algorithm

Cosine is a graph coloring algorithm which has been especially designed for producing optimal colorings in a large class of perfect graphs. This algorithm is used as a heuristic method for getting an upper bound on the chromatic number of general graphs; this estimation is then considered as an initial step of a more sophisticated coloring procedure based on tabu search techniques. Cosine is compared with other methods having similar complexity and we show that Cosine realizes a good compromise between computational time and performance level.

Short-chorded and perfect graphs

A graph is short-chroded (a.k.a. Raspail) if every odd cycle of length at least 5 has a short chord, which is a chord joining vertices distance 2 apart in the cycle. A subclass of short-chorded graphs, not contained in any of the known classes of perfect graphs, will be proved perfect.

Efficient algorithms for computing the reliability of permutation and interval graphs

AbstractA stochastic network in which nodes fail randomly with known probabilities is modeled by a probabilistic graph with unreliable nodes and perfect edges. The K‐terminal reliability of such a network is the probability that there exists a Steiner tree connecting a subset of the nodes K (target nodes). Although the K‐terminal reliability problem has been widely studied for networks with unreliable links, very little is known about the problem for networks with unreliable nodes. We show that computing this measure is computationally difficult, in particular #P‐complete. We then present efficient algorithms for the K‐terminal reliability problem on two classes of perfect graphs; interval graphs and permutation graphs. Computing the reliability on these two classes of graphs is of particular interest since the problem remains #P‐complete for larger classes in the hierarchy of perfect graphs, namely, comparability and chordal graphs. The model presented in this paper is appropriate for radio broadcast networks and for fault‐tolerant multiprocessor networks.

Perfectly contractile graphs

We define a new hereditary class of perfect graphs G, the perfectly contractile graphs, which have the following property: Any γ( G)-colouring of G can be obtained from any k-colouring of G by a sequence of switchings. Then we show that Gavril's clique separable graphs (hence also i-triangulated graphs), Chvátal's perfectly orderable graphs (hence also cotriangulated and comparability graphs), and Meyniel graphs (hence also parity graphs) are perfectly contractile.

An nc algorithm to recognize hhd-free graphs

The HHD-free graphs are a class of perfect graphs that strictly contain the cographs, the chordal graphs, the Matula-perfect graphs, and the Welsh-Powell opposition graphs. In this paper we present two characterizations of the HHD-free graphs and show how to use them to produce a parallel recognition algorithm which runs in O(log n) time using O(n 4) processors on a concurrent-read, concurrent-write parallel random access machine.

Some classes of perfectly orderable graphs

AbstractIn 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.