Separator Theorems for Interval Graphs and Proper Interval Graphs

k-tuple domination in graphs

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.

We introduce a new subclass of chordal graphs that generalizes the class of split graphs, which we call well-partitioned chordal graphs. A connected graph G is a well-partitioned chordal graph if there exist a partition P of the vertex set of G into cliques and a tree T having P as a vertex set such that for distinct X,Y∈P, (1) the edges between X and Y in G form a complete bipartite subgraph whose parts are some subsets of X and Y, if X and Y are adjacent in T, and (2) there are no edges between X and Y in G otherwise. A split graph with vertex partition (C,I) where C is a clique and I is an independent set is a well-partitioned chordal graph as witnessed by a star T having C as the center and each vertex in I as a leaf, viewed as a clique of size 1. We characterize well-partitioned chordal graphs by forbidden induced subgraphs, and give a polynomial-time algorithm that given a graph, either finds an obstruction, or outputs a partition of its vertex set that asserts that the graph is well-partitioned chordal.We observe that there are problems, for instance Densestk-Subgraph and b-Coloring, that are polynomial-time solvable on split graphs but become NP -hard on well-partitioned chordal graphs. On the other hand, we show that the Geodetic Set problem, known to be NP -hard on chordal graphs, can be solved in polynomial time on well-partitioned chordal graphs.We also answer two combinatorial questions on well-partitioned chordal graphs that are open on chordal graphs, namely that each well-partitioned chordal graph admits a polynomial-time constructible tree 3-spanner, and that each (2-connected) well-partitioned chordal graph has a vertex that intersects all its longest paths (cycles).

Some properties of edge intersection graphs of single-bend paths on a grid

Corneil, Olariu, and Stewart [SODA 1998; SIAM Journal on Discrete Mathematics 2009] presented a recognition algorithm for interval graphs by six graph searches. Li and Wu [Discrete Mathematics \& Theoretical Computer Science 2014] simplified it to only four. The great simplicity of the latter algorithm is however eclipsed by the complicated and long proofs. The main purpose of this paper is to present a new and significantly short proof for Li and Wu's algorithm, as well as a simpler implementation. We also give a self-contained simpler interpretation of the recognition algorithm of Corneil [Discrete Applied Mathematics 2004] for unit interval graphs, based on three sweeps of graph searches. Moreover, we show that two sweeps are already sufficient. Toward the proofs of the main results, we make several new structural observations that might be of independent interests.

For a simple, undirected graph [Formula: see text], a function [Formula: see text] which satisfies the following conditions is called an outer-independent total Roman dominating function (OITRDF) of [Formula: see text] with weight [Formula: see text]. (C1) For all [Formula: see text] with [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], (C2) The induced subgraph with vertex set [Formula: see text] has no isolated vertices and (C3) The induced subgraph with vertex set [Formula: see text] is independent. For a graph [Formula: see text], the smallest possible weight of an OITRDF of [Formula: see text] which is denoted by [Formula: see text], is known as the outer-independent total Roman domination number of [Formula: see text]. The problem of determining [Formula: see text] of a graph [Formula: see text] is called minimum outer-independent total Roman domination problem (MOITRDP). In this article, we show that the problem of deciding if [Formula: see text] has an OITRDF of weight at most [Formula: see text] for bipartite graphs and split graphs, a subclass of chordal graphs is NP-complete. We also show that MOITRDP is linear time solvable for connected threshold graphs and bounded treewidth graphs. Finally, we show that the domination and outer-independent total Roman domination problems are not equivalent in computational complexity aspects.

Computing role assignments of split graphs

A set of vertices $S$ of a graph $G$ is {\em monophonically convex} if every induced path joining two vertices of $S$ is contained in $S$. The {\em monophonic convex hull of $S$}, $\langle S \rangle$, is the smallest monophonically convex set containing $S$. A set $S$ is {\em monophonic convexly independent} if $v \not\in \langle S - \{v\} \rangle$ for every $v \in S$. The {\em monophonic rank} of $G$ is the size of the largest monophonic convexly independent set of $G$. We present a characterization of the monophonic convexly independent sets. Using this result, we show how to determine the monophonic rank of graph classes like bipartite, cactus, triangle-free, and line graphs in polynomial time. Furthermore, we show that this parameter can computed in polynomial time for $1$-starlike graphs, i.e., for split graphs, and that its determination is $\NP$-complete for $k$-starlike graphs for any fixed $k \ge 2$, a subclass of chordal graphs. We also consider this problem on the graphs whose intersection graph of the maximal prime subgraphs is a tree.

Minimal proper interval completions

The flooding games, which are called Flood-It, Mad Virus, or HoneyBee, are a kind of coloring games and they have been becoming popular online. In these games, each player colors one specified cell in his/her turn, and all connected neighbor cells of the same color are also colored by the color. This flooding or coloring spreads on the same color cells. It is natural to consider these new coloring games on more general boards, or general graphs. Recently, computational complexities of the variants of the flooding games on several graph classes have been studied. In this paper, we investigate the flooding games on some graph classes characterized by interval representations. Our results state that the number of colors is a key parameter to determine the computational complexity of the flooding games. When the number of colors is a fixed constant, these games can be solved in polynomial time on an interval graph. On the other hand, if the number of colors is not bounded, the flooding game is NP-complete on a proper interval graph. We also state similar results for split graphs.

We study interval coloring problems and present new upper and lower bounds for several variants. We are interested in four problems, online coloring of intervals with and without bandwidth and a new problem called lazy online coloring again with and without bandwidth. We consider both general interval graphs and unit interval graphs. Specifically, we establish the difference between the two main problems which are interval coloring with and without bandwidth. We present the first non-trivial lower bound of 3.2609 for the problem with bandwidth. This improves the lower bound of 3 that follows from the tight results for interval coloring without bandwidth presented in [9].

A polynomial algorithm for the k-cluster problem on the interval graphs

Edge integrity of nearest neighbor graphs and separator theorems

Normal Helly circular-arc graphs and its subclasses

Let G be a connected simple graph. We prove that G is a closed graph if and only if G is a proper interval graph. As a consequence we obtain that there exist linear-time algorithms for closed graph recognition.