Clique-Based Separators for Geometric Intersection Graphs

Let F be a set of n objects in the plane and let G^x(F) be its intersection graph. A balanced clique-based separator of G^x(F) is a set S consisting of cliques whose removal partitions G^x(F) into components of size at most δn, for some fixed constant δ < 1. The weight of a clique-based separator is defined as ∑_{C ∈ S} log (|C|+1). Recently De Berg et al. (SICOMP 2020) proved that if S consists of convex fat objects, then G^x(F) admits a balanced clique-based separator of weight O(√n). We extend this result in several directions, obtaining the following results. - Map graphs admit a balanced clique-based separator of weight O(√n), which is tight in the worst case. - Intersection graphs of pseudo-disks admit a balanced clique-based separator of weight O(n^{2/3} log n). If the pseudo-disks are polygonal and of total complexity O(n) then the weight of the separator improves to O(√n log n). - Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight O(n^{2/3} log n). - Visibility-restricted unit-disk graphs in a polygonal domain with r reflex vertices admit a balanced clique-based separator of weight O(√n + r log(n/r)), which is tight in the worst case. These results immediately imply sub-exponential algorithms for MAXIMUM INDEPENDENT SET (and, hence, VERTEX COVER), for FEEDBACK VERTEX SET, and for q-Coloring for constant q in these graph classes.ISSN:1868-896

On the chromatic number of multiple interval graphs and overlap graphs

AbstractLet χ(G) and ω(G) denote the chromatic number and clique number of a graph G. We prove that χ can be bounded by a function of ω for two well-known relatives of interval graphs. Multiple interval graphs (the intersection graphs of sets which can be written as the union of t closed intervals of a line) satisfy χ⩽2t(ω−1) for ω⩾2. Overlap graphs satisfy χ⩽2ωω2(ω−1)

On 3-Coloring Circle Graphs

Given a graph $G$ with a fixed vertex order $\prec$, one obtains a circle graph $H$ whose vertices are the edges of $G$ and where two such edges are adjacent if and only if their endpoints are pairwise distinct and alternate in $\prec$. Therefore, the problem of determining whether $G$ has a $k$-page book embedding with spine order $\prec$ is equivalent to deciding whether $H$ can be colored with $k$ colors. Finding a $k$-coloring for a circle graph is known to be NP-complete for $k \geq 4$ and trivial for $k \leq 2$. For $k = 3$, Unger (1992) claims an efficient algorithm that finds a 3-coloring in $O(n \log n)$ time, if it exists. Given a circle graph $H$, Unger's algorithm (1) constructs a 3-\textsc{Sat} formula $\Phi$ that is satisfiable if and only if $H$ admits a 3-coloring and (2) solves $\Phi$ by a backtracking strategy that relies on the structure imposed by the circle graph. However, the extended abstract misses several details and Unger refers to his PhD thesis (in German) for details. In this paper we argue that Unger's algorithm for 3-coloring circle graphs is not correct and that 3-coloring circle graphs should be considered as an open problem. We show that step (1) of Unger's algorithm is incorrect by exhibiting a circle graph whose formula $\Phi$ is satisfiable but that is not 3-colorable. We further show that Unger's backtracking strategy for solving $\Phi$ in step (2) may produce incorrect results and give empirical evidence that it exhibits a runtime behaviour that is not consistent with the claimed running time.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Minimizing the number of vehicles to meet a fixed periodic schedule : an application of periodic posets

Includes bibliographical references (leaf 25).Supported in part by the U.S. Army Research Office. DAAG29-80-C-0061by James B. Orlin

On the minimum and maximum selective graph coloring problems in some graph classes

Given a graph together with a partition of its vertex set, the minimum selective coloring problem consists of selecting one vertex per partition set such that the chromatic number of the subgraph induced by the selected vertices is minimum. The contribution of this paper is twofold. First, we investigate the complexity status of the minimum selective coloring problem in some specific graph classes motivated by some models described in Demange et al. (2015). Second, we introduce a new problem that corresponds to the worst situation in the minimum selective coloring; the maximum selective coloring problem aims to select one vertex per partition set such that the chromatic number of the subgraph induced by the selected vertices is maximum. We motivat

The permutation-path coloring problem on trees

AbstractIn this paper we first show that the permutation-path coloring problem is NP-hard even for very restrictive instances like involutions, which are permutations that contain only cycles of length at most two, on both binary trees and on trees having only two vertices with degree greater than two, and for circular permutations, which are permutations that contain exactly one cycle, on trees with maximum degree greater than or equal to 4. We calculate a lower bound on the average complexity of the permutation-path coloring problem on arbitrary networks. Then we give combinatorial and asymptotic results for the permutation-path coloring problem on linear networks in order to show that the average number of colors needed to color any permutation on a linear network on n vertices is n/4+o(n). We extend these results and obtain an upper bound on the average complexity of the permutation-path coloring problem on arbitrary trees, obtaining exact results in the case of generalized star trees. Finally we explain how to extend these results for the involutions-path coloring problem on arbitrary trees