New Class Of Graphs Research Articles

Completely positive matrices with a book-graph

A new class of graphs, called “book-graphs”, extending the class of completely positive graphs is defined. Necessary and sufficient conditions for the complete positivity of a matrix with graph in this class are given. The main questions concerning completely positive matrices with cyclic graph are solved.

Triangulating multitolerance graphs

In this paper we introduce a new class of graphs which generalize both the tolerance and the trapezoid graphs, the multitolerance graphs. We show that the difference between the pathwidth and the treewidth of a multitolerance graph is at most one, and we develop an algorithm which solves the minimum fill-in problem for a multitolerance graph with a given representation in polynomial time. These results complement the recent results of Habib and Möhring [18, 25] about the treewidth and pathwidth of cocomparability graphs and graphs without asteroidal triples, and those of Kloks et al. [21] about the minimum fill-in problem.

On the structure of graphs with few P4s

We present new classes of graphs for which the isomorphism problem can be solved in polynomial time. These graphs are characterized by containing — in some local sense — only a small number of induced paths of length three. As it turns out, every such graph has a unique tree representation: the internal nodes correspond to three types of graph operations, while the leaves are basic graphs with a simple structure. The paper extends and generalizes known results about cographs, P 4-reducible graphs, and P 4-sparse graphs.

Global insertion and hamiltonicity in DCT-graphs

In a new class of graphs strictly containing the class of almost claw-free graphs, the class of quasi claw-free graphs and thus the class of claw-free graphs, we show that if the graph G is 2-connected of order n and if the degree sum of any three independent vertices is at least n − 2, then G is hamiltonian. This problem was posed for almost claw-free graphs by Broersma et al. (1996) and was settled by Li and Tian when n ⩾ 79 for another class containing the almost claw-free graphs. We also consider properties of matchings and toughness in this new class. In the main proof, we introduce a technique of global insertion which is a more powerful tool than the usual insertion.

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.

Trapezoid graphs and generalizations, geometry and algorithms

Trapezoid graphs are a class of cocomparability graphs containing interval graphs and permutation graphs as subclasses. They were introduced by Dagan et al. [3]. They propose an O( n 2) algorithm for chromatic number and a less efficient algorithm for maximum clique on trapezoid graphs. Based on a geometric representation of trapezoid graphs by boxes in the plane we design optimal, i.e., O( n log n), algorithms for chromatic number, weighted independent set, clique cover and maximum weighted clique on such graphs. We also propose generalizations of trapezoid graphs called k-trapezoid graphs. The ideas behind the clique cover and weighted independent set algorithms for trapezoid graphs carry over to higher dimensions. This leads to O( n log k−1 n) algorithms for k-trapezoid graphs. We also propose a new class of graphs called circle trapezoid graphs. This class contains trapezoid graphs, circle graphs and circular-arc graphs as subclasses. We show that clique and independent set problems for circle trapezoid graphs are efficiently solvable. The algorithms solving these two problems require algorithms for trapezoid graphs as subroutines.

On semi- P4-sparse graphs

In this paper, we define a graph G as semi- P 4-sparse if G does not contain as induced subgraph a P 5, a P 5 or the complement of a fork, where a fork is the tree of order 5 with 3 pendent vertices. This new class of graphs contains strictly the class of P 4-sparse graphs. Using modular decomposition we first propose a linear recognition algorithm for semi- P 4-sparse graphs and next, we show that with very little work, we can extend the linear algorithms of Chvátal et al. (1987) concerning the class of perfect graphs that are P 5, P 5 and C 5-free, for finding an optimal coloring, a largest clique and a largest stable set of a semi- P 4-sparse graph G. Finally, we characterize by forbidden configurations the closure by substitution-composition (the inverse operation of modular decomposition) of semi- P 4-sparse graphs, composition operation of graphs that (among other properties) preserves perfection.

On weakly diamond-free Berge graphs

In this paper, we present a new class of graphs named weakly diamond-free (WDF) graphs and we prove for it the strong perfect graph conjecture, by exhibiting a polynomial sequential ω-coloring algorithm. This class contains chordal graphs and perfect line-graphs.

On powers of m-trapezoid graphs

First a new class of graphs is introduced: m-trapezoid graphs are the intersection graphs of m-trapezoid, where an m-trapezoid is given by m + 1 intervals on m + 1 parallel lines. The main result of this paper is that if G k − 1 is an m-trapezoid graph then G k is also an m-trapezoid graph. This theorem has some interesting corollaries concerning interval graphs, trapezoid graphs and cocomparability graphs: If A is either of these classes, then G k − 1 ϵ A implies G k ϵ A . This answers an open question of P. Damaschke.

Competitive inheritance and limitedness of graphs

AbstractThe study of competition number of graphs has brought many interesting problems to graph theory and combinatorics. However, little is known about what makes the competition number of a graph small or large. In this paper, we introduce competitive inheritance and limitedness of graphs. We show that most graphs studied in the literature of competition graphs possess the inheritance property on their competition number. We characterize the limitedness of graphs on competition number by forbidden induced subgraphs. Our results also prove a conjecture of Opsut on competition number for a new class of graphs. © 1995 John Wiley & Sons, Inc.

Homomorphism and sigma polynomials

By establishing a connection between the sigma polynomial and the homomorphism polynomial, many of the proofs for computing the sigma polynmial are simplified, the homomorphism polynomial can be identified for several new classes of graphs, and progress can be made on identifying homomorphism polynomials.

On the stability number of AH‐free graphs

AbstractWe describe a new class of graphs for which the stability number can be obtained in polynomial time. The algorithm is based on an iterative procedure that, at each step, builds from a graph G a new graph Gl that has fewer nodes and has the property that α(Gl) = α(G) − 1.This new class of graphs is different from the known classes for which the stability number can be computed in polynomial time. © 1993 John Wiley & Sons, Inc.

The strong perfect graph conjecture holds for diamonded odd cycle-free graphs

In this paper, we show that the strong perfect graph conjecture holds for a new class of graphs which we call diamonded odd cycle-free graphs. The class of diamonded odd cycle-free graphs contains the classes of threshold graphs and K 4\\ e-free graphs.

On graphs with equal domination and independent domination numbers

Allan and Laskar have shown that K 1.3-free graphs with equal domination and independent domination numbers. In this paper new classes of graphs with equal domination independent domination numbers.are presented. In particular, the result of Allan and Laskar is generalized.

Unified Theory of Database Serializability12

A database system is a collection of data items, read or written by transactions in a possibly interleaved fashion. An interleaved execution is assumed to be correct if the sequence of the steps of the transactions, called history, is serializable, that is, the effect of the execution is equivalent to that of some serial execution of the same transactions. In this paper we give a new characterization of serializability that brings out the inherent problem of serialization explicitly. We then give a graph-theoretic analogue of serializable histories. We define a new class of graphs, called serializable graphs, whose properties are such that (i) a serializable graph can be associated with each serializable history, and this can be done for various notions of serializability of histories and for serializability under various sets of constraints, and (ii) a serializable history, in fact a serial one, can be associated with each serializable graph. We use serializable graphs to characterize, in an intuitive manner, serializable histories involving general multi-step transactions, where the same data item can be accessed by several read and write steps of a transaction in an arbitrary manner, and those involving nested transactions. We also define a new notion of serializability for nested transactions. This enables relating several acceptable concurrent executions of transactions, that are not serializable with the traditional transaction concept, to sequential behaviour. Serializability under this notion is also characterized. The main graph-theoretic properties used in these characterizations are a directed cutset matching property and graph contraction.

Bipolarizable graphs

We consider all perfectly orderable graphs which admit a bipolarized perfect order. This new class of graphs will be characterized by a collection of forbidden induced subgraphs. We shall compare this class with other well known classes of perfectly orderable graphs.

B-banyan and B-delta networks for multiprocessor systems

A new class of graphs, B-banyan, and a new class of multistage interconnection networks based on B-banyans, B-delta networks, in multiprocessor environments are proposed in this paper. A B-banyan is a regular rectangular SW-banyan augmented with a backward link at each vertex, in a way that preserves the destination control. A backward link can be taken when there is a conflict at a node, or when a faulty link or node is encountered. As such, backward links in B-delta networks significantly increase the network performance while at the same time providing a degree of fault tolerance. The performance of B-delta networks is analyzed and shown to be comparable to that of single-buffered delta networks. Different switches and control complexities of these two types of delta networks are discussed.

P4‐Reducible Graphs—Class of Uniquely Tree‐Representable Graphs

We introduce and characterize a new class of graphs which has a unique tree representation, and which strictly contains the well‐known class of cographs. We define three graph operations and show that all the graphs in our class can be constructed from single‐vertex graphs by a finite sequence of these operations. Finally, we show that a number of computational problems, including the four classical optimization problems, can be solved efficiently for this new class of graphs.

On the Maximum Weight Clique Problem

We introduce several new classes of graphs on which the maximum-weight clique problem is solvable in polynomial time. Their common feature, and the central idea of our algorithms, is that every clique of any of our graphs is contained in some member of a polynomial-sized collection of induced subgraphs that are complements of bipartite graphs.

Feedback vertex sets and cyclically reducible graphs

The problem of finding a minimum cardinality feedback vertex set of a directed graph is considered. Of the classic NP-complete problems, this is one of the least understood. Although Karp showed the general problem to be NP-complete, a linear algorithm for its solution on reducible flow graphs was given by Shamir. The class of reducible flow graphs is the only nontrivial class of graphs for which a polynomial-time algorithm to solve this problem is known. The main result of this paper is to present a new class of graphs—the cyclically reducible graphs —for which minimum feedback vertex sets can be found in polynomial time. This class is not restricted to flow graphs, and most small graphs (10 or fewer nodes) fall into this class. The identification of this class is particularly important since there do not exist approximation algorithms for this problem having a provably good worst case performance. Along with the class and a simple polynomial-time algorithm for finding minimum feedback vertex sets of graphs in the class, several related results are presented. It is shown that there is no “forbidden subgraph” characterization of the class and that there is no particular inclusion relationship between this class and the reducible flow graphs. In addition, it is shown that a class of (general) graphs, which are related to the reducible flow graphs, are contained in the cyclically reducible class.