On minimal forbidden subgraph characterizations of balanced graphs

A graph is balanced if its clique-matrix contains no edge–vertex incidence matrix of an odd chordless cycle as a submatrix. While a forbidden induced subgraph characterization of balanced graphs is known, there is no such characterization by minimal forbidden induced subgraphs. In this work, we provide minimal forbidden induced subgraph characterizations of balanced graphs restricted to graphs that belong to one of the following graph classes: complements of bipartite graphs, line graphs of multigraphs, and complements of line graphs of multigraphs. These characterizations lead to linear-time recognition algorithms for balanced graphs within the same three graph classes.     

Forbidden ordered subgraph vs. forbidden subgraph characterizations of graph classes

We show that the minimum set of unordered graphs that must be forbidden to get the same graph class characterized by forbidding a single ordered graph is infinite. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 71–76, 1999     

A forbidden subgraph characterization of line-polar bipartite graphs

A graph is polar if the vertex set can be partitioned into A and B in such a way that the subgraph induced by A is a complete multipartite graph and the subgraph induced by B is a disjoint union of cliques. Polar graphs are a common generalization of bipartite, cobipartite, and split graphs. However, recognizing polar graphs is an NP-complete problem in general. This led to the study of the polarity of special classes of graphs such as cographs and chordal graphs, cf. Ekim et al. (2008) [7] and [5]. In this paper, we study the polarity of line graphs and call a graph line-polar if its line graph is polar. We characterize line-polar bipartite graphs in terms of forbidden subgraphs. This answers a question raised in the fist reference mentioned above. Our characterization has already been used to develop a linear time algorithm for recognizing line-polar bipartite graphs, cf. Ekim (submitted for publication) [6].     

Four forbidden subgraph pairs for hamiltonicity of 3-connected graphs

Acta Mathematicae Applicatae Sinica, English Series - For non-negative integers i, j and k, we denote the generalized net as N i,j,k , which is a triangle with disjoint paths of length i, j and k,...     

Universal graphs with a forbidden subgraph: Block path solidity

Let C be a finite connected graph for which there is a countable universal C-free graph, and whose tree of blocks is a path. Then the blocks of C are complete. This generalizes a result of Füredi and Komjáth, and fits naturally into a set of conjectures regarding the existence of countable C-free graphs, with C an arbitrary finite connected graph.     

On the forbidden induced subgraph sandwich problem

We consider the sandwich problem, a generalization of the recognition problem introduced by Golumbic et al. (1995) [15], with respect to classes of graphs defined by excluding induced subgraphs. We prove that the sandwich problem corresponding to excluding a chordless cycle of fixed length k is NP-complete. We prove that the sandwich problem corresponding to excluding K_r?e for fixed r is polynomial. We prove that the sandwich problem corresponding to 3PC(⋅,⋅)-free graphs is NP-complete. These complexity results are related to the classification of a long-standing open problem: the sandwich problem corresponding to perfect graphs.     

Forbidden induced subgraph characterizations of subclasses and variations of perfect graphs: A survey

A graph is perfect if the chromatic number is equal to the clique number for every induced subgraph of the graph. Perfect graphs were defined by Berge in the sixties. In this survey we present known results about partial characterizations by forbidden induced subgraphs of different graph classes related to perfect graphs. We analyze a variation of perfect graphs, clique-perfect graphs, and two subclasses of perfect graphs, coordinated graphs and balanced graphs.     

Searching for a minimal solution subgraph in explicit AND/OR graphs

An algorithm for searching for a minimal solution subgraph in AND/OR graphs with cycles is described, which works top–down and is appropriate to explicit AND/OR graphs.     

ON THE ASCENDING SUBGRAPH DECOMPOSITIONS OF REGULAR GRAPHS

The definition of the ascending suhgraph decomposition was given by Alavi. It has been conjectured that every graph of positive size has an ascending subgraph decomposition. In this paper it is proved that the regular graphs under some conditions do have an ascending subgraph decomposition.     

On planar intersection graphs with forbidden subgraphs

Let be a family of n compact connected sets in the plane, whose intersection graph has no complete bipartite subgraph with k vertices in each of its classes. Then has at most n times a polylogarithmic number of edges, where the exponent of the logarithmic factor depends on k. In the case where consists of convex sets, we improve this bound to O(n log n). If in addition k = 2, the bound can be further improved to O(n). © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 205–214, 2008     

On double bound graphs and forbidden subposets

For a poset P=(X,≤_P), the double bound graph (DB-graph) of P is the graph DB(P)=(X,E_DB(P)), where xy∈E_DB(P) if and only if x≠y and there exist n,m∈X such that n≤_Px,y≤_Pm. We obtain that for a subposet Q of a poset P,Q is an (n, m)-subposet of P if and only if DB(Q) is an induced subgraph DB(P). Using this result, we show some characterizations of split double bound graphs, threshold double bound graphs and difference double bound graphs in terms of (n, m)-subposets and double canonical posets.     

On automorphisms of infinite graphs with forbidden subgraphs

LexX be anm-connected infinite graph without subgraphs homeomorphic toKm, n, for somen, and let α be an automorphism ofX with at least one cycle of infinite length. We characterize the structure of α and use this characterization to extend a known result about orientation-preserving automorphisms of finite plane graphs to infinite plane graphs. In the last section we investigate the action of α on the ends ofX and show that α fixes at most two ends (Theorem 3.2).     

On the balanced domination of graphs

Czechoslovak Mathematical Journal - Let G = (VG,EG) be a graph and let NG[υ] denote the closed neighbourhood of a vertex υ in G. A function f: VG ? {?1,0,1} is said to be a...     

On spectral characterizations and embeddings of graphs

The least eigenvalue of the 0-1 adjacency matrix of a graph is denoted λ G. In this paper all graphs with λ(G) greater than ?2 are characterized. Such a graph is a generalized line graph of the form L(T;1,0,…,0), L(T), L(H), where T is a tree and H is unicyclic with an odd cycle, or is one of 573 graphs that arise from the root system E₈. If G is regular with λ(G)>?2, then Gis a clique or an odd circuit. These characterizations are used for embedding problems; λ_R(H) = sup{λ(G)z.sfnc;HinG; Gregular}. H is an odd circuit, a path, or a complete graph iff λ_R(H)> ?2. For any other line graph H, λ_R(H) = ?2. A similar result holds for complete multipartite graphs.     

On the generation of circuits and minimal forbidden sets

We present several complexity results related to generation and counting of all circuits of an independence system. Our motivation to study these problems is their relevance in the solution of resource constrained scheduling problems, where an independence system arises as the subsets of jobs that may be scheduled simultaneously. We are interested in the circuits of this system, the so-called minimal forbidden sets, which are minimal subsets of jobs that must not be scheduled simultaneously. As a consequence of the complexity results for general independence systems, we obtain several complexity results in the context of resource constrained scheduling. On that account, we propose and analyze a simple backtracking algorithm that generates all minimal forbidden sets for such problems. The performance of this algorithm, in comparison to a previously suggested divide-and-conquer approach, is evaluated empirically using instances from the project scheduling library PSPLIB.Acknowledgement We appreciate the input of two anonymous referees. Particularly the deep remarks of one of them greatly improved our understanding of several issues; he also suggested the simplified Example 1. We thank Marc Pfetsch and Alexander Grigoriev for several enlightening discussions. Marc Pfetsch also pointed us to the paper [15]. Parts of this work were done while the authors were PhD students at the Technische Universität Berlin, Germany, where Frederik Stork was supported by DFG grant Mo 446/3-3, and Marc Uetz was supported by bmb+f grant 03-MO7TU1-3 and GIF grant I 246-304.02/97.     

On classes of minimal circular-imperfect graphs

Circular-perfect graphs form a natural superclass of perfect graphs: on the one hand due to their definition by means of a more general coloring concept, on the other hand as an important class of χ-bound graphs with the smallest non-trivial χ-binding function χ(G)?ω(G)+1.The Strong Perfect Graph Conjecture, recently settled by Chudnovsky et al. [The strong perfect graph theorem, Ann. of Math. 164 (2006) 51-229], provides a characterization of perfect graphs by means of forbidden subgraphs. It is, therefore, natural to ask for an analogous conjecture for circular-perfect graphs, that is for a characterization of all minimal circular-imperfect graphs.At present, not many minimal circular-imperfect graphs are known. This paper studies the circular-(im)perfection of some families of graphs: normalized circular cliques, partitionable graphs, planar graphs, and complete joins. We thereby exhibit classes of minimal circular-imperfect graphs, namely, certain partitionable webs, a subclass of planar graphs, and odd wheels and odd antiwheels. As those classes appear to be very different from a structural point of view, we infer that formulating an appropriate conjecture for circular-perfect graphs, as analogue to the Strong Perfect Graph Theorem, seems to be difficult.     

On bipartite graphs with minimal energy

The energy of a graph is the sum of the absolute values of the eigenvalues of the graph. In a paper [G. Caporossi, D. Cvetkovi, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with external energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996] Caporossi et al. conjectured that among all connected graphs G with n≥6 vertices and n−1≤m≤2(n−2) edges, the graphs with minimum energy are the star S_n with m−n+1 additional edges all connected to the same vertices for m≤n+⌊(n−7)/2⌋, and the bipartite graph with two vertices on one side, one of which is connected to all vertices on the other side, otherwise. The conjecture is proved to be true for m=n−1,2(n−2) in the same paper by Caporossi et al. themselves, and for m=n by Hou in [Y. Hou, Unicyclic graphs with minimal energy, J. Math. Chem. 29 (2001) 163-168]. In this paper, we give a complete solution for the second part of the conjecture on bipartite graphs. Moreover, we determine the graph with the second-minimal energy in all connected bipartite graphs with n vertices and edges.