An on-line competitive algorithm for coloring bipartite graphs without long induced paths

The existence of an on-line competitive algorithm for coloring bipartite graphs remains a tantalizing open problem. So far there are only partial positive results for bipartite graphs with certain small forbidden graphs as induced subgraphs. We propose a new on-line competitive coloring algorithm for $P_9$-free bipartite graphs

Graph classes equivalent to 12-representable graphs

Jones et al. (2015) introduced the notion of $u$-representable graphs, where $u$ is a word over $\{1, 2\}$ different from $22\cdots2$, as a generalization of word-representable graphs. Kitaev (2016) showed that if $u$ is of length at least 3, then every graph is $u$-representable. This indicates that there are only two nontrivial classes in the theory of $u$-representable graphs: 11-representable graphs, which correspond to word-representable graphs, and 12-representable graphs. This study deals with 12-representable graphs. Jones et al. (2015) provided a characterization of 12-representable trees in terms of forbidden induced subgraphs. Chen and Kitaev (2022) presented a forbidden induced subgraph characterization of a subclass of 12-representable grid graphs. This paper shows that a bipartite graph is 12-representable if and only if it is an interval containment bigraph. The equivalence gives us a forbidden induced subgraph characterization of 12-representable bipartite graphs since the list of minimal forbidden induced subgraphs is known for interval containment bigraphs. We then have a forbidden induced subgraph characterization for grid graphs, which solves an open problem of Chen and Kitaev (2022). The study also shows that a graph is 12-representable if and only if it is the complement of a simple-triangle graph. This equivalence indicates that a necessary condition for 12-representability presented by Jones et al. (2015) is also sufficient. Finally, we show from these equivalences that 12-representability can be determined in $O(n^2)$ time for bipartite graphs and in $O(n(\bar{m}+n))$ time for arbitrary graphs, where $n$ and $\bar{m}$ are the number of vertices and edges of the complement of the given graph.Comment: 12 pages, 6 figure

On Forbidden Induced Subgraphs for Unit Disk Graphs

A unit disk graph is the intersection graph of disks of equal radii in the plane. The class of unit disk graphs is hereditary, and therefore admits a characterization in terms of minimal forbidden induced subgraphs. In spite of quite active study of unit disk graphs very little is known about minimal forbidden induced subgraphs for this class. We found only finitely many minimal non-unit disk graphs in the literature. In this paper we study in a systematic way forbidden induced subgraphs for the class of unit disk graphs. We develop several structural and geometrical tools, and use them to reveal infinitely many new minimal non-unit disk graphs. Further we use these results to investigate structure of co-bipartite unit disk graphs. In particular, we give structural characterization of those co-bipartite unit disk graphs whose edges between parts form a C4-free bipartite graph, and show that bipartite complements of these graphs are also unit disk graphs. Our results lead us to propose a conjecture that the class of co-bipartite unit disk graphs is closed under bipartite complementation

Density theorems for bipartite graphs and related Ramsey-type results

In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniques can be used to study properties of graphs with a forbidden induced subgraph, edge intersection patterns in topological graphs, and to obtain several other Ramsey-type statements

Boundary properties of graphs

A set of graphs may acquire various desirable properties, if we apply suitable restrictions on the set. We investigate the following two questions: How far, exactly, must one restrict the structure of a graph to obtain a certain interesting property? What kind of tools are helpful to classify sets of graphs into those which satisfy a property and those that do not? Equipped with a containment relation, a graph class is a special example of a partially ordered set. We introduce the notion of a boundary ideal as a generalisation of a notion introduced by Alekseev in 2003, to provide a tool to indicate whether a partially ordered set satisfies a desirable property or not. This tool can give a complete characterisation of lower ideals defined by a finite forbidden set, into those that satisfy the given property and to those that do not. In the case of graphs, a lower ideal with respect to the induced subgraph relation is known as a hereditary graph class. We study three interrelated types of properties for hereditary graph classes: the existence of an efficient solution to an algorithmic graph problem, the boundedness of the graph parameter known as clique-width, and well-quasi-orderability by the induced subgraph relation. It was shown by Courcelle, Makowsky and Rotics in 2000 that, for a graph class, boundedness of clique-width immediately implies an efficient solution to a wide range of algorithmic problems. This serves as one of the motivations to study clique-width. As for well-quasiorderability, we conjecture that every hereditary graph class that is well-quasi-ordered by the induced subgraph relation also has bounded clique-width. We discover the first boundary classes for several algorithmic graph problems, including the Hamiltonian cycle problem. We also give polynomial-time algorithms for the dominating induced matching problem, for some restricted graph classes. After discussing the special importance of bipartite graphs in the study of clique-width, we describe a general framework for constructing bipartite graphs of large clique-width. As a consequence, we find a new minimal class of unbounded clique-width. We prove numerous positive and negative results regarding the well-quasi-orderability of classes of bipartite graphs. This completes a characterisation of the well-quasi-orderability of all classes of bipartite graphs defined by one forbidden induced bipartite subgraph. We also make considerable progress in characterising general graph classes defined by two forbidden induced subgraphs, reducing the task to a small finite number of open cases. Finally, we show that, in general, for hereditary graph classes defined by a forbidden set of bounded finite size, a similar reduction is not usually possible, but the number of boundary classes to determine well-quasi-orderability is nevertheless finite. Our results, together with the notion of boundary ideals, are also relevant for the study of other partially ordered sets in mathematics, such as permutations ordered by the pattern containment relation