Chordless Path Research Articles

Linear time recognition of P4-indifference graphs

A graph is a P4-indifference graph if it admits an ordering < on its vertices such that every chordless path with vertices a, b, c, d and edges ab, bc, cd has a

On the P4-components of graphs

Two edges are called P 4-adjacent if they belong to the same P 4 (chordless path on four vertices). P 4-components, in our terminology, are the equivalence classes of the transitive closure of the P 4-adjacency relation. In this paper, new results on the structure of P 4-components are obtained. On the one hand, these results allow us to improve the complexity of orienting P 4-comparability graphs and of recognizing P 4-indifference graphs from O( n 5) and O( n 6) to O( m 2). On the other hand, by combining the modular decomposition with the substitution of P 4-components, a new unique tree representation for arbitrary graphs is derived which generalizes the homogeneous decomposition introduced by Jamison and Olariu (SIAM J. Discrete Math. 8 (1995) 448–463).

Linear Time Recognition of P4-Indifferent Graphs

<p>A simple graph is P4-indifferent if it admits a total order < on<br />its nodes such that every chordless path with nodes a, b, c, d and edges<br />ab, bc, cd has a < b < c < d or a > b > c > d. P4-indifferent graphs generalize<br /> indifferent graphs and are perfectly orderable. Recently, Hoang,<br />Maray and Noy gave a characterization of P4-indifferent graphs in<br />terms of forbidden induced subgraphs. We clarify their proof and describe<br /> a linear time algorithm to recognize P4-indifferent graphs. When<br />the input is a P4-indifferent graph, then the algorithm computes an order < as above.</p><p>Key words: P4-indifference, linear time, recognition, modular decomposition.</p><p> </p>

A characterization ofP4-indifference graphs

A graph is a P4-indifference graph if it admits a linear ordering ≺ on its vertices such that every chordless path with vertices a, b, c, d and edges ab, bc, cd has either a ≺ b ≺ c ≺ d or d ≺ c ≺ b ≺ a. P4-indifference graphs generalize indifference graphs and are perfectly orderable. We give a characterization of P4-indifference graphs by forbidden induced subgraphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 155-162, 1999

On the p-connectedness of graphs – a survey

A graph is said to be p-connected if for every partition of its vertices into two non-empty, disjoint, sets some chordless path with three edges contains vertices from both sets in the partition. As it turns out, p-connectedness generalizes the usual connectedness of graphs and leads, in a natural way, to a unique tree representation for arbitrary graphs. This paper reviews old and new results, both structural and algorithmic, about p-connectedness along with applications to various graph decompositions.

Tree- and forest-perfect graphs

Two graphs G and H with the same vertex set V are P 4- isomorphic if there exists a permutation π on V such that, for all subsets S⊆V, S induces a chordless path on four vertices (denoted by P 4) in G if and only if π( S) induces a P 4 in H. This paper gives a classification of all graphs P 4-isomorphic to a tree, respectively, a forest.

Bipartite-Perfect Graphs

Abstract Two graphs G and H with the same vertex set V are P4-isomorphic if there exists a permutation π on V such that, for all subset S ⊆V, S induces a chordless path on four vertices (denoted by P4)in G if and only if π(S) induces a P4 in H.This paper gives a classification of all graphs P4-isomorphic to a bipartite graph, which we call bipartite-perfect graphs. The classification is based on graphs P4-isomorphic to a tree previously described by A. Brandstadt and the author, and yields a linear time recognition algorithm for bipartite-perfect graphs.

Even Pairs in Claw-Free Perfect Graphs

An even pair in a graph is a pair of non-adjacent vertices such that every chordless path between them has even length. A graph is called strict quasi-parity when every induced subgraph that is not a clique has an even pair, and it is called perfectly contractile when every induced subgraph can be turned into a clique through a sequence of even-pair contractions. In this paper we determine the K 1, 3 -free graphs that are strict quasi-parity and those that are perfectly contractile. We show that for both classes the minimal forbidden configurations are odd holes, antiholes and some line-graphs of bipartite graphs, as conjectured by several authors. Our proofs are constructive and yield polynomial-time algorithms for the recognition of both classes.

Tree-like P4-connected graphs

A graph is P 4-connected if for every partition of its vertices into two nonempty disjoint sets there is a chordless path on four vertices which contains vertices from both sets in the partition. An alternative characterization states that a graph is P 4-connected if and only if any two vertices are connected by a P 4-chain, that is, a sequence of vertices such that every four consecutive ones induce a P 4. In this paper we study graphs where each induced subgraph contains a vertex which belongs to at most one P 4. It turns out that the P 4-connected components of these graphs are provided with structural properties which can be expressed in a quite analogous way to the numerous characterizations of ordinary trees. Among others, we present characterizations by forbidden subgraphs, in terms of the number of P 4s, and by the uniqueness of P 4-chains connecting two vertices.

A fast parallel algorithm to recognize P4-sparse graphs

A number of problems in mobile computing, group-based collaboration, automated theorem proving, networking, scheduling, and cluster analysis suggested the study of graphs featuring certain “local density” characteristics. Typically, the notion of local density is equated with the absence of chordless paths of length three or more. Recently, a new metric for local density has been proposed, allowing a number of such induced paths to occur. More precisely, a graph G is called P4-sparse if no set of five vertices in G induces more than one chordless path of length three. P4-sparse graphs generalize the well-known class of cographs corresponding to a more stringent local density metric. One remarkable feature of P4-sparse graphs is that they admit a tree representation unique up to isomorphism. In this work we present a parallel algorithm to recognize P4-sparse graphs and show how the data structures returned by the recognition algorithm can be used to construct the corresponding tree representation. With a graph G= (V, E) with¦V¦= n and¦E¦= m as input, our algorithms run in O(logn) time using O((n 2 + mn)/ log n) processors in the EREW-PRAM model.

On Planar Perfectly Contractile Graphs

An even pair in a graph is a pair of vertices such that every chordless path between them has even length. A graph is called perfectly contractile when every induced subgraph can be transformed into a clique through a sequence of even-pair contractions. In this paper we characterize the planar graphs that are perfectly contractile by determining all the minimal forbidden subgraphs. We give a polynomial algorithm for the recognition of perfectly contractile planar graphs.

On extended P4-reducible and extended P4-sparse graphs

A graph G was defined in [16] as P 4-reducible , if no vertex in G belongs to more than one chordless path on four vertices or P 4. A graph G is defined in [15] as P 4-sparse if no set of five vertices induces more than one P 4, in G. P 4-sparse graphs generalize both P 4-reducible and the well known class of p 4-free graphs or cographs. In an extended abstract in [11] the first author introduced a method using the modular decomposition tree of a graph as the framework for the resolution of algorithmic problems. This method was applied to the study of P 4-sparse and extended P 4-sparse graphs. In this paper, we begin by presenting the complete information about the method used in [11]. We propose a unique tree representation of P 4-sparse and a unique tree representation of P 4-reducible graphs leading to a simple linear recognition algorithm for both classes of graphs. In this way we simplify and unify the solutions for these problems, presented in [16–19]. The tree representation of an n-vertex P 4-sparse or a P 4-reducible graph is the key for obtaining O( n) time algorithms for the weighted version of classical optimization problems solved in [20]. These problems are NP-complete on general graphs. Finally, by relaxing the restriction concerning the exclusion of the C 5 cycles from P 4-sparse and P 4-reducible graphs, we introduce the class of the extended P4-sparse and the class of the extended P 4-reducible graphs. We then show that a minimal amount of additional work suffices for extending most of our algorithms to these new classes of graphs.

Properly orderable graphs

In a graph G = ( V, E) provided with a linear order ‘ < ’ on V, a chordless path with vertices a, b, c, d and edges ab, bc, cd is called an obstruction if both a < b and d < c hold. Chvátal (1984) defined the class of perfectly orderable graphs (i.e., graphs possessing an acyclic orientation of the edges such that no obstruction is induced) and proved that they are perfect. We introduce here the class of properly orderable graphs which is a generalization of Chvátal's class of perfectly orderable graphs: obstructions are forbidden only in the subgraphs induced by the vertices of an odd cycle. We prove the perfection of these graphs and give an O( m 2 + mn + n) colouring algorithm.

A polynomial algorithm for the parity path problem on perfectly orientable graphs

The parity path problem is to test if a graph contains an even (odd) chordless path between two prescribed vertices. A graph is called perfectly orientable if there exists an orientation of its edges such that the resulting digraph has no obstructions: an obstruction is a chordless path ( a, b, c, d) having →ab and ←cd. We present a polynomial algorithm for the parity path problem on perfectly orientable graphs.

RecognizingP3-Structure: A Switching Approach

A P 3 is a set of three vertices of a graph that induces a chordless path; the P 3 -structure of a graph is the set of all P 3 's. Chvátal asked if there is a polynomial time algorithm to determine whether an arbitrary three-uniform hypergraph is the P 3 -structure of some graph. The answer is yes: we describe such an algorithm. Our algorithm is based on switching (as defined by Seidel), a characterization of graphs with the same P 3 -structure, a sufficient condition for P 3 -structure uniqueness, and a reduction of a certain partitioning problem to two-satisfiability.

A linear-time recognition algorithm for P4-reducible graphs

The P 4-reducible graphs are a natural generalization of the well-known class of cographs, with applications to scheduling, computational semantics, and clustering. More precisely, the P 4-reducible graphs are exactly the graphs none of whose vertices belong to more than one chordless path with three edges. A remarkable property of P 4-reducible graphs is their unique tree representation up to isomorphism. In this paper we present a linear-time algorithm to recognize P 4-reducible graphs and to construct their corresponding tree representation.

Linear time optimization algorithms for P4-sparse graphs

Quite often, real-life applications suggest the study of graphs that feature some local density properties. In particular, graphs that are unlikely to have more than a few chordless paths of length three appear in a number of contexts. A graph G is P 4-sparse if no set of five vertices in G induces more than one chordless path of length three. P 4-sparse graphs generalize both the class of cographs and the class of P 4-reducible graphs. It has been shown that P 4-sparse graphs can be recognized in time linear in the size of the graph. The main contribution of this paper is to show that once the data structures returned by the recognition algorithm are in place, a number of NP-hard problems on general graphs can be solved in linear time for P 4-sparse graphs. Specifically with an n-vertex P 4-sparse graph as input the problems of finding a maximum size clique, maximum size stable set, a minimum coloring, a minimum covering by clique, and the size of the minimum fill-in can be solved in O( n) time, independent of the number of edges in the graph.

On the two‐edge‐colorings of perfect graphs

AbstractWe investigate the conjecture that a graph is perfect if it admits a two‐edge‐coloring such that two edges receive different colors if they are the nonincident edges of a P4 (chordless path with four vertices). Partial results on this conjecture are given in this paper. © 1995 John Wiley &amp; Sons, Inc.

Which claw-free graphs are perfectly orderable?

An undirected graph is called perfectly orderable if the set of its vertices admits a linear order < such that no chordless path with vertices a, b, c, d and edges ab, bc, cd has a<b and d<c. We characterize claw-free perfectly orderable graphs by nine infinite families of forbidden induced subgraphs. This result generalizes the characterization of totally balanced matrices found by Anstee and Farber, Edmonds and Lubiw, and Hoffman, Kolen and Sakarovitch. Implicit in our argument is a polynomial-time algorithm that, given any claw-free graph G, either constructs a perfect order in G or finds one of the forbidden induced subgraphs in G.

A linear algorithm for the group path problem on chordal graphs

Assume that each edge of a graph G=( V, E) is given a weight, which is an element of some group G . The weight of a path P is defined as the product of the weights of the edges along P. The group path problem is to find a chordless path of a given weight between two given vertices. It generalizes the parity path problem considered by Hsu. We show that the recognition problem associated with the group path problem is NP-complete in general, and present an O(| G|·|E|+|V|) time algorithm for the group path problem on a chordal graph.