A characterization of cycle-free unit probe interval graphs

A graph is a probe interval graph (PIG) if its vertices can be partitioned into probes and nonprobes with an interval assigned to each vertex so that vertices are adjacent if and only if their corresponding intervals overlap and at least one of them is a probe. PIGs are a generalization of interval graphs introduced by Zhang for an application concerning the physical mapping of DNA in the human genome project. PIGs have been characterized in the cycle-free case by Sheng, and other miscellaneous results are given by McMorris, Wang, and Zhang. Johnson and Spinrad give a polynomial time recognition algorithm for when the partition of vertices into probes and nonprobes is given. The complexity for the general recognition problem is not known. Here, we restrict attention to the case where all intervals have the same length, that is, we study the unit probe interval graphs and characterize the cycle-free graphs that are unit probe interval graphs via a list of forbidden induced subgraphs.     

The complexity of dissociation set problems in graphs

A subset of vertices in a graph is called a dissociation set if it induces a subgraph with a vertex degree of at most 1. The maximum dissociation set problem, i.e., the problem of finding a dissociation set of maximum size in a given graph is known to be NP-hard for bipartite graphs. We show that the maximum dissociation set problem is NP-hard for planar line graphs of planar bipartite graphs. In addition, we describe several polynomially solvable cases for the problem under consideration. One of them deals with the subclass of the so-called chair-free graphs. Furthermore, the related problem of finding a maximal (by inclusion) dissociation set of minimum size in a given graph is studied, and NP-hardness results for this problem, namely for weakly chordal and bipartite graphs, are derived. Finally, we provide inapproximability results for the dissociation set problems mentioned above.     

Adjacency matrices of probe interval graphs

In this paper we obtain several characterizations of the adjacency matrix of a probe interval graph. In course of this study we describe an easy method of obtaining interval representation of an interval bigraph from its adjacency matrix. Finally, we note that if we add a loop at every probe vertex of a probe interval graph, then the Ferrers dimension of the corresponding symmetric bipartite graph is at most 3.     

Almost all 5-regular graphs have a 3-flow

Tutte conjectured in 1972 that every 4-edge–connected graph has a nowhere-zero 3-flow. This has long been known to be equivalent to the conjecture that every 5-regular 4-edge–connected graph has an edge orientation in which every in-degree is either 1 or 4. We show that the assertion of the conjecture holds asymptotically almost surely for random 5-regular graphs. It follows that the conjecture holds for almost all 4-edge–connected 5-regular graphs.     

Large 2P3-free graphs with bounded degree

Let ex^* (D;H) be the maximum number of edges in a connected graph with maximum degree D and no induced subgraph H; this is finite if and only if H is a disjoint union of paths. If the largest component of such an H has order m, then ex^*(D; H) = O(D²ex^*(D; P_m)). Constructively, ex^*(D;qP_m) = Θ(gD²ex^*(D;P_m)) if q>1 and m> 2(Θ(gD²) if m = 2). For H = 2P₃ (and D 8), the maximum number of edges is if D is even and

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if D is odd, achieved by a unique extremal graph.     

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].     

Claw-free graphs with strongly perfect complements. Fractional and integral version. Part I. Basic graphs

Strongly perfect graphs have been studied by several authors (e.g. Berge and Duchet (1984) [1], Ravindra (1984) [12] and Wang (2006) [14]). In a series of two papers, the current paper being the first one, we investigate a fractional relaxation of strong perfection. Motivated by a wireless networking problem, we consider claw-free graphs that are fractionally strongly perfect in the complement. We obtain a forbidden induced subgraph characterization and display graph-theoretic properties of such graphs. It turns out that the forbidden induced subgraphs that characterize claw-free graphs that are fractionally strongly perfect in the complement are precisely the cycle of length 6, all cycles of length at least 8, four particular graphs, and a collection of graphs that are constructed by taking two graphs, each a copy of one of three particular graphs, and joining them in a certain way by a path of arbitrary length. Wang (2006) [14] gave a characterization of strongly perfect claw-free graphs. As a corollary of the results in this paper, we obtain a characterization of claw-free graphs whose complements are strongly perfect.     

Chordal graphs,interval graphs,and wqo

Let precedes, equal to be the induced-minor relation. It is shown that, for every t, all chordal graphs of clique number at most t are well-quasi-ordered by precedes, equal to. On the other hand, if the bound on clique number is dropped, even the class of interval graphs is not well-quasi-ordered by precedes, equal to. © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 105–114, 1998     

Edge intersection graphs of linear 3-uniform hypergraphs

Let be the class of edge intersection graphs of linear 3-uniform hypergraphs. It is known that the problem of recognition of the class is NP-complete. We prove that this problem is polynomially solvable in the class of graphs with minimum vertex degree ≥10. It is also proved that the class is characterized by a finite list of forbidden induced subgraphs in the class of graphs with minimum vertex degree ≥16.     

A Characterization of Domination Reducible Graphs

We introduce a hereditary class of domination reducible graphs where the minimum dominating set problem is polynomially solvable, and characterize this class in terms of forbidden induced subgraphs.Acknowledgments The research was supported by DIMACS 2002 and 2003 Awards. The author would like to thank the both referees for their valuable suggestions.Final version received: October 3, 2003     

Mock threshold graphs

Mock threshold graphs are a simple generalization of threshold graphs that, like threshold graphs, are perfect graphs. Our main theorem is a characterization of mock threshold graphs by forbidden induced subgraphs. Other theorems characterize mock threshold graphs that are claw-free and that are line graphs. We also discuss relations with chordality and well-quasi-ordering as well as algorithmic aspects.     

Polarity of chordal graphs

Polar graphs are a common generalization of bipartite, cobipartite, and split graphs. They are defined by the existence of a certain partition of vertices, which is NP-complete to decide for general graphs. It has been recently proved that for cographs, the existence of such a partition can be characterized by finitely many forbidden subgraphs, and hence tested in polynomial time. In this paper we address the question of polarity of chordal graphs, arguing that this is in essence a question of colourability, and hence chordal graphs are a natural restriction. We observe that there is no finite forbidden subgraph characterization of polarity in chordal graphs; nevertheless we present a polynomial time algorithm for polarity of chordal graphs. We focus on a special case of polarity (called monopolarity) which turns out to be the central concept for our algorithms. For the case of monopolar graphs, we illustrate the structure of all minimal obstructions; it turns out that they can all be described by a certain graph grammar, permitting our monopolarity algorithm to be cast as a certifying algorithm.     

All vertex-transitive locally-quasiprimitive graphs have a semiregular automorphism

The polycirculant conjecture states that every transitive 2-closed permutation group of degree at least two contains a nonidentity semiregular element, that is, a nontrivial permutation whose cycles all have the same length. This would imply that every vertex-transitive digraph with at least two vertices has a nonidentity semiregular automorphism. In this paper we make substantial progress on the polycirculant conjecture by proving that every vertex-transitive, locally-quasiprimitive graph has a nonidentity semiregular automorphism. The main ingredient of the proof is the determination of all biquasiprimitive permutation groups with no nontrivial semiregular elements. M. Giudici is supported by an Australian Postdoctoral Fellowship J. Xu was supported by an IPRS scholarship of Australia.     

Online Ramsey theory for a triangle on -free graphs

Given a class of graphs and a fixed graph , the online Ramsey game for H on is a game between two players Builder and Painter as follows: an unbounded set of vertices is given as an initial state, and on each turn Builder introduces a new edge with the constraint that the resulting graph must be in , and Painter colors the new edge either red or blue. Builder wins the game if Painter is forced to make a monochromatic copy of at some point in the game. Otherwise, Painter can avoid creating a monochromatic copy of forever, and we say Painter wins the game. We initiate the study of characterizing the graphs such that for a given graph , Painter wins the online Ramsey game for on -free graphs. We characterize all graphs such that Painter wins the online Ramsey game for on the class of -free graphs, except when is one particular graph. We also show that Painter wins the online Ramsey game for on the class of -minor-free graphs, extending a result by Grytczuk, Hałuszczak, and Kierstead [Electron. J. Combin. 11 (2004), p. 60].     

Note on 2-edge-colorings of complete graphs with small monochromatic k-connected subgraphs

Bollob′as and Gy′arf′as conjectured that for n4(k-1) every 2-edge-coloring of Kn contains a monochromatic k-connected subgraph with at least n-2k+2 verticesLiu, et alproved that the conjecture holds when n≥13k-15In this note, we characterize all the2-edge-colorings of Kn where each monochromatic k-connected subgraph has at most n-2k+2 vertices for n≥13k-15.