On edge transitivity of directed graphs

Word-representability of graphs with respect to split recomposition

On polygon numbers of circle graphs and distance hereditary graphs

: This paper introduces a class of 1 player games of perfect information, which we call complementing games; the player is allowed moves which complement the value of successive plays. A complementing game is symmetric if all noncomplement moves are reversible (i.e., form a symmetric relation). These games are naturally related to a class of machines we call symmetric complementing machines. Symmetric nondeterministic machines were studied in (Lewis and Papadimitriou, 801): they are identical to our symmetric complementing machines with complement moves allowed only on termination. (A companion paper to appear will describe the computational complexity of symmetric complementing and alternating machines.) Of particular interest is the complexity class Sum *CSYMLOG, which contains the outcome problem of symmetric complementing games with constant complement bound with game positions encoded in log space, and next move relations computable in log space. We show that the decision problem for a restricted quantified Boolean logic Sum *QBF sum is complete in Sum *CSYMLOG. We also show that Sum *CSYMLOG contains many well- known and common combinatiorial problems: minimum spanning forests; k- connectivity and k-connected components, and also the recognition problems for many classes of graphs: planar graphs of valence 3; chordal graphs; comparability graphs; interval graphs; split graphs, and permutation graphs.

Treelike Comparability Graphs

Consider a class of graphs $$\mathcal{G}$$ having a polynomial time algorithm computing the set of all minimal separators for every graph in $$\mathcal{G}$$ . We show that there is a polynomial time algorithm for treewidth and minimum fill-in, respectively, when restricted to the class $$\mathcal{G}$$ . Many interesting classes of intersection graphs have a polynomial time algorithm computing all minimal separators, like permutation graphs, circle graphs, circular arc graphs, distance hereditary graphs, chordal bipartite graphs etc. Our result generalizes earlier results for the treewidth and minimum fill-in for several of these classes. We also consider the related problems pathwidth and interval completion when restricted to some special graph classes.

Graph isomorphism completeness for chordal bipartite graphs and strongly chordal graphs

Chapter 11 Not so perfect graphs

The last chapter of this book eventually presents some famous classes of perfect graphs, namely comparability, interval, permutation, and split graphs. We first present an example of a graph which is at the same time a triangulated, a comparability, a split, and a permutation graph. The importance to be an interval is illustrated with Berge’s mystery story. We discuss furthermore the generation of permutation graphs and close with how to recognise that a given graph is in fact a permutation graph.

Klavík et al. [arXiv:1207.6960] recently introduced a generalization of recognition called the bounded representation problem which we study for the classes of interval and proper interval graphs. The input gives a graph G and in addition for each vertex v two intervals \(\mathfrak{L}_v\) and \(\mathfrak{R}_v\) called bounds. We ask whether there exists a bounded representation in which each interval \(\mathfrak{I}_v\) has its left endpoint in \(\mathfrak{L}_v\) and its right endpoint in \(\mathfrak{R}_v\). We show that the problem can be solved in linear time for interval graphs and in quadratic time for proper interval graphs.Robert’s Theorem states that the classes of proper interval graphs and unit interval graphs are equal. Surprisingly, the bounded representation problem is polynomially solvable for proper interval graphs and NP-complete for unit interval graphs [Klavík et al., arxiv:1207.6960]. So unless P = NP, the proper and unit interval representations behave very differently.The bounded representation problem belongs to a wider class of restricted representation problems. These problems are generalizations of the well-understood recognition problem, and they ask whether there exists a representation of G satisfying some additional constraints. The bounded representation problems generalize many of these problems.

Comparability graph augmentation for some multiprocessor scheduling problems

String graphs of k-bend paths on a grid

The maximum independent set problem (MaxIS) on general graphs is known to be NP-hard to approximate within a factor of $n^{1-epsilon}$, for any $epsilon > 0$. However, there are many ``easy classes of graphs on which the problem can be solved in polynomial time. In this context, an interesting question is that of computing the maximum independent set in a graph that can be expressed as the union of a small number of graphs from an easy class. The MaxIS problem has been studied on unions of interval graphs and chordal graphs. We study the MaxIS problem on unions of perfect graphs (which generalize the above two classes). We present an $O(sqrt{n})$-approximation algorithm when the input graph is the union of two perfect graphs. We also show that the MaxIS problem on unions of two comparability graphs (a subclass of perfect graphs) cannot be approximated within any constant factor.

Given a probabilistic graph, with reliable edges and unreliable vertices, $K$ -terminal reliability problem is to find the probability that a given subset $K$ of vertices remains connected. The problem is #P-complete for general graphs, even so for some special graphs such as chordal graphs, and comparability graphs. However, polynomial time algorithms to solve the problem have been designed for some special graphs such as interval graphs, permutation graphs, and $d$ -trapezoid graphs. Existing time complexity of the polynomial time algorithm for the $d$ -trapezoid graph is $O(n^{2d+1})$ , which is a very high degree polynomial. This makes it impractical to solve large problem instances using the existing algorithm. Here, we propose a novel technique and use it to design a simple linear-time algorithm to solve the problem on $d$ -trapezoid graphs. As the complexity of our algorithm is linear, it is no more a difficulty to solve large problem instances in small amount of time.

On Semi-Transitive Orientability of Split Graphs

We consider graph properties that can be checked from labels, i.e., bit sequences, of logarithmic length attached to vertices. We prove that there exists such a labeling for checking a first-order formula with free set variables in the graphs of every class that is \emph{nicely locally cwd-decomposable}. This notion generalizes that of a \emph{nicely locally tree-decomposable} class. The graphs of such classes can be covered by graphs of bounded \emph{clique-width} with limited overlaps. We also consider such labelings for \emph{bounded} first-order formulas on graph classes of \emph{bounded expansion}. Some of these results are extended to counting queries.