New Class Of Graphs Research Articles

Polar graphs and maximal independent sets

We study two central problems of algorithmic graph theory: finding maximum and minimum maximal independent sets. Both problems are known to be NP-hard in general. Moreover, they remain NP-hard in many special classes of graphs. For instance, the problem of finding minimum maximal independent sets has been recently proven to be NP-hard in the class of so-called ( 1 , 2 ) -polar graphs. On the other hand, both problems can be solved in polynomial time for ( 1 , 1 ) -polar, also known as split graphs. In this paper, we address the question of distinguishing new classes of graphs admitting polynomial-time solutions for the two problems in question. To this end, we extend the hierarchy of ( α , β ) -polar graphs and study the computational complexity of the problems on polar graphs of special types.

Classes of perfect graphs

The strong perfect graph conjecture, suggested by Claude Berge in 1960, had a major impact on the development of graph theory over the last 40 years. It has led to the definitions and study of many new classes of graphs for which the strong perfect graph conjecture has been verified. Powerful concepts and methods have been developed to prove the strong perfect graph conjecture for these special cases. In this paper we survey 120 of these classes, list their fundamental algorithmic properties and present all known relations between them.

A general approach to L(h, k)-label interconnection networks

Given two non-negative integers h and k, an L(h, k)-labeling of a graph G = (V, E) is a function from the set V to a set of colors, such that adjacent nodes take colors at distance at least h, and nodes at distance 2 take colors at distance at least k. The aim of the L(h, k)-labeling problem is to minimize the greatest used color. Since the decisional version of this problem is NP-complete, it is important to investigate particular classes of graphs for which the problem can be efficiently solved. It is well known that the most common interconnection topologies, such as Butterfly-like, Benes, CCC, Trivalent Cayley networks, are all characterized by a similar structure: they have nodes organized as a matrix and connections are divided into layers. So we naturally introduce a new class of graphs, called (l × n)-multistage graphs, containing the most common interconnection topologies, on which we study the L(h, k)-labeling. A general algorithm for L(h, k)-labeling these graphs is presented, and from this method an efficient L(2, 1)-labeling for Butterfly and CCC networks is derived. Finally we describe a possible generalization of our approach.

A linear time algorithm for the minimum Weighted Feedback Vertex Set on diamonds

Given an undirected and vertex weighted graph G, the Weighted Feedback Vertex Problem (WFVP) consists in finding a subset F ⊆ V of vertices of minimum weight such that each cycle in G contains at least one vertex in F. The WFVP on general graphs is known to be NP-hard. In this paper we introduce a new class of graphs, namely the diamond graphs, and give a linear time algorithm to solve WFVP on it.

Minimum Weighted Feedback Vertex Set on Diamonds

Given a vertex weighted graph G, a minimum Weighted Feedback Vertex Set (MWFVS) is a subset F µ V of vertices of minimum weight such that each cycle in G contains at least one vertex in F. The MWFVS on general graph is known to be NP-hard. In this paper we introduce a new class of graphs, namely the diamond graphs, and give a linear time algorithm to solve MWFVS on it. We will discuss, moreover, how this result could be used to efiectively improve the approximated solution of any known heuristic to solve MWFVS on a general graph.

Growth of 1-Periodic Graphs

A new class of graphs is introduced: 1-periodic graphs. A problem on the form of the growth of these graphs is stated. For certain classes of graphs, a self-similar growth in the form of a polygon is proved. Certain general conjectures on the form of the growth are stated. Bibliography: 1 title.

A polynomial algorithm for finding T-span of generalized cacti

It has been known for years that the problem of computing the T-span is NP-hard in general. Recently, Giaro et al. (Discrete Appl. Math., to appear) showed that the problem remains NP-hard even for graphs of degree Δ⩽3 and it is polynomially solvable for graphs with degree Δ⩽2. Herein, we extend the latter result. We introduce a new class of graphs which is large enough to contain paths, cycles, trees, cacti, polygon trees and connected outerplanar graphs. Next, we study the properties of graphs from this class and prove that the problem of computing the T-span for these graphs is polynomially solvable.

On the quasi-locally paw-free graphs

In this paper, we present a new class of graphs named quasi-locally paw-free (QLP) graphs. We prove the strong perfect graph conjecture for a subclass of QLP class, by exhibiting a polynomial combinatorial algorithm for ω-coloring any Berge graph for this subclass. This subclass contains K 4-free graphs and chordal graphs.

Pk+1–Decompositions of Eulerian Graphs: Complexity and Some Solvable Cases

We consider the problem of PMk+1-decomposition of a simple eulerian graph G, that is, decomposition of G into edge disjoint paths of length k. We show that the problem of deciding whether there exists a Pk+1 - decomposition of an eulerian simple graph is NP-complete, for every k = 3. However we find some new classes of graphs where the problem of P4-decomposition can be solved polynomially. We show that an eulerian simple graph G on 3m = 6 edges admits a P4-decomposition if G has no cut vertex v such that exactly one of the components in the graph G - ? has two vertices. In particular, this implies that a 2-connected eulerian simple graph G on 3m = 6 edges admits a P4 -decomposition. © 2003.

Bi‐arc graphs and the complexity of list homomorphisms

AbstractGiven graphs G, H, and lists L(v) ⊆ V(H), v ε V(G), a list homomorphism of G to H with respect to the lists L is a mapping f : V(G) → V(H) such that uv ε E(G) implies f(u)f(v) ε E(H), and f(v) ε L(v) for all v ε V(G). The list homomorphism problem for a fixed graph H asks whether or not an input graph G, together with lists L(v) ⊆ V(H), v ε V(G), admits a list homomorphism with respect to L. In two earlier papers, we classified the complexity of the list homomorphism problem in two important special cases: When H is a reflexive graph (every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NP‐complete otherwise. When H is an irreflexive graph (no vertex has a loop), the problem is polynomial time solvable if H is bipartite and H is a circular arc graph, and is NP‐complete otherwise. In this paper, we extend these classifications to arbitrary graphs H (each vertex may or may not have a loop). We introduce a new class of graphs, called bi‐arc graphs, which contains both reflexive interval graphs (and no other reflexive graphs), and bipartite graphs with circular arc complements (and no other irreflexive graphs). We show that the problem is polynomial time solvable when H is a bi‐arc graph, and is NP‐complete otherwise. In the case when H is a tree (with loops allowed), we give a simpler algorithm based on a structural characterization. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 61–80, 2003

A Transformation Which Preserves the Clique Number

We introduce a graph transformation which preserves the clique number. When applied to graphs containing no odd hole and no cricket (a particular graph on 5 vertices) the transformation also preserves the chromatic number. Using this transformation we derive a polynomial algorithm for the computation of the clique number of all graphs in a class which strictly contains diamond-free graphs. Furthermore, the transformation leads to a proof that the Strong Perfect Graph Conjecture is true for two new classes of graphs and yields a polynomial time algorithm for the computation of the clique number and the chromatic number for both classes. One of these two classes strictly contains claw-free graphs.

Depender graphs: A method of fault-tolerant certificate distribution1

We consider scalable certificate revocation in a public-key infrastructure (PKI). We introduce depender graphs, a new class of graphs that support efficient and fault-tolerant revocation. Nodes of a depender graph are participants that agree to forwa

Forestal algebras and algebraic forests (on a new class of weakly compact graphs)

In this paper we introduce and investigate a new class of graphs called algebraic forests for which isomorphism testing can be done in time O(n 3 log n) . The class of algebraic forests admits a membership test of the same complexity, it includes cographs, trees and interval graphs, and even a joint superclass of the latter two, namely, rooted-directed path graphs. In fact, our class is much larger than these classes, since every graph is an induced subgraph of some algebraic forest. The key point of our approach is the study of the class of forestal cellular algebras defined inductively from one-point algebras by taking direct sums and wreath products. In fact, algebraic forests are exactly the graphs the cellular algebras of which are forestal. We prove that each weak isomorphism of two forestal algebras is induced by a strong isomorphism. This implies that all forestal algebras are compact cellular algebras and so all algebraic forests are weakly compact graphs. We also present a complete description of cellular algebras of disconnected graphs.

On computing the minimum feedback vertex set of a directed graph by contraction operations

Finding the minimum feedback vertex set (MFVS) in a graph is an important problem for a variety of computer-aided design (CAD) applications and graph reduction plays an important role in solving this intractable problem. This paper is largely concerned with three new and powerful reduction operations. Each of these operations defines a new class of graphs, strictly larger than the class of contractible graphs [Levy and Low (1988)] in which the MFVS can be found in polynomial-time complexity. Based on these operations, an exact algorithm run on branch and bound manner is developed. This exact algorithm uses a good heuristic to find out an initial solution and a good bounding strategy to prune the solution space. To demonstrate the efficiency of our algorithms, we have implemented our algorithms and applied them to solving the partial scan problem in ISCAS'89 benchmarks. The experimental results show that if our three new contraction operations are applied, 27 out of 31 circuits in ISCAS'89 benchmarks can be fully reduced. Otherwise, only 12 out of 31 can be fully reduced. Furthermore, for all ISCAS'89 benchmarks our exact algorithm can find the exact cutsets in less than 3 s (CPU time) on a SUN-UltraII workstation. Therefore, the new contraction operations and our algorithms are demonstrated to be very effective in the partial scan application.

The game coloring number of pseudo partial k-trees

This paper introduces a new class of graphs: ( a, b)-pseudo partial k-trees. In some sense, the parameters ( a, b) measure a graph's deviation from being a partial k-tree. In particular, a (0,0)-pseudo partial k-tree is just a partial k-tree. We discuss the game coloring number (as well as the game chromatic number) of ( a, b)-pseudo partial k-trees, and prove that the game coloring number of an ( a, b)-pseudo partial k-tree is at most 3 k+2 a+ b+2. In particular, the game coloring number of a partial k-tree is at most 3 k+2. This reduces considerably the previous known upper bound for the game chromatic number of a partial k-tree. Then we describe another strategy for Alice for playing the game, which gives a better upper bound for the game coloring number of ( a, b)-pseudo partial 2-trees. Namely, we prove that the game coloring number of an ( a, b)-pseudo partial 2-tree is at most a+ b+8. By using this result, we prove that the game coloring number of a graph embeddable on an orientable surface of genus g⩾1 is at most ⌊ 1 2 (3 1+48g +23)⌋. This is the first upper bound for the game coloring number of such graphs, and it also improves considerably the previous known upper bound for the game chromatic number of such graphs. Moreover, the strategy for Alice introduced in this paper for playing the game can be considered as a common generalization of the previous known strategies for playing the game on forests, interval graphs, outerplanar graphs and planar graphs.

Convex-Round and Concave-Round Graphs

We introduce two new classes of graphs which we call convex-round, respectively concave-round graphs. Convex-round (concave-round) graphs are those graphs whose vertices can be circularly enumerated so that the (closed) neighborhood of each vertex forms an interval in the enumeration. Hence the two classes transform into each other by taking complements. We show that both classes of graphs have nice structural properties. We observe that the class of concave-round graphs properly contains the class of proper circular arc graphs and, by a result of Tucker [ Pacific J. Math., 39 (1971), pp. 535--545], is properly contained in the class of general circular arc graphs. We point out that convex-round and concave-round graphs can be recognized in O(n+m) time (here n denotes the number of vertices and m the number of edges of the graph in question). We show that the chromatic number of a graph which is convex-round (concave-round) can be found in time O(n+m) (O(n2 )). We describe optimal O(n+m) time algorithms for finding a maximum clique, a maximum matching, and a Hamiltonian cycle (if one exists) for the class of convex-round graphs. Finally, we pose a number of open problems and conjectures concerning the structure and algorithmic properties of the two new classes and a related third class of graphs.

Berge graphs with chordless cycles of bounded length

A graph is called weakly triangulated if it contains no chordless cycle on five or more vertices (also called hole) and no complement of such a cycle (also called antihole). Equivalently, we can define weakly triangulated graphs as antihole-free graphs whose induced cycles are isomorphic either to C3 or to C4. The perfection of weakly triangulated graphs was proved by Hayward [Hayward, J Combin Theory B. 39 (1985), 200–208] and generated intense studies to efficiently solve, for these graphs, the classical NP-complete problems that become polynomial on perfect graphs. If we replace, in the definition above, the C4 by an arbitrary Cp (p even, at least equal to 6), we obtain new classes of graphs whose perfection is shown in this article. In fact, we prove a more general result: for any even integer p ≥ 6, the graphs whose cycles are isomorphic either to C3 or to one of Cp, Cp+2, …, C2p 6 are perfect. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 73–79, 1999

Berge graphs with chordless cycles of bounded length

A graph is called weakly triangulated if it contains no chordless cycle on five or more vertices (also called hole) and no complement of such a cycle (also called antihole). Equivalently, we can define weakly triangulated graphs as antihole-free graphs whose induced cycles are isomorphic either to C3 or to C4. The perfection of weakly triangulated graphs was proved by Hayward [Hayward, J Combin Theory B. 39 (1985), 200–208] and generated intense studies to efficiently solve, for these graphs, the classical NP-complete problems that become polynomial on perfect graphs. If we replace, in the definition above, the C4 by an arbitrary Cp (p even, at least equal to 6), we obtain new classes of graphs whose perfection is shown in this article. In fact, we prove a more general result: for any even integer p ≥ 6, the graphs whose cycles are isomorphic either to C3 or to one of Cp, Cp+2, …, C2p 6 are perfect. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 73–79, 1999

Enumerative aspects of certain subclasses of perfect graphs

We investigate the enumerative aspects of various classes of perfect graphs like cographs, split graphs, trivially perfect graphs and threshold graphs. For subclasses of permutation graphs like cographs and threshold graphs we also determine the number of permutations π of {1,2,…, n} such that the permutation graph G[ π] belongs to that class. We establish an interesting bijection between permutations whose permutation graphs are cographs ( P 4-free graphs) and permutations that are obtainable using an output-restricted deque (Knuth, Art of Computer Programming, Vol I, Fundamental Algorithms) and thereby enumerate such permutations. We also prove that the asymptotic number of permutations of {1,2,…, n} whose permutation graphs are split graphs is Θ(4 n/ n ) . We also introduce a new class of graphs called C 5-split graphs, characterize and enumerate them. C 5-split graphs form a superclass of split graphs and are not necessarily perfect. All the classes of graphs that we enumerate have a finite family of small forbidden induced subgraphs.

On minimal imperfect graphs without induced P5

In this paper, we summarize previous known results on P 5-free minimal imperfect graphs (i.e. minimal imperfect graphs not containing a path on 5 vertices as induced subgraph) and we introduce two new classes of graphs (defined by a local property) that contain P 5-free graphs. Next, we show that most of the results concerning P 5-free graphs can be extended to these classes. Moreover, we present a structural characterization of these graphs which leads to some new results. In particular, we prove that the Strong Perfect Graph Conjecture holds true for P 5-free and F-free graphs where F is any connected configuration on 5 vertices not containing an induced 2 K 2.