25 research outputs found
Planar graph coloring avoiding monochromatic subgraphs: trees and paths make things difficult
We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem
Induced Minor Free Graphs: Isomorphism and Clique-width
Given two graphs and , we say that contains as an induced
minor if a graph isomorphic to can be obtained from by a sequence of
vertex deletions and edge contractions. We study the complexity of Graph
Isomorphism on graphs that exclude a fixed graph as an induced minor. More
precisely, we determine for every graph that Graph Isomorphism is
polynomial-time solvable on -induced-minor-free graphs or that it is
GI-complete. Additionally, we classify those graphs for which
-induced-minor-free graphs have bounded clique-width. These two results
complement similar dichotomies for graphs that exclude a fixed graph as an
induced subgraph, minor, or subgraph.Comment: 16 pages, 5 figures. An extended abstract of this paper previously
appeared in the proceedings of the 41st International Workshop on
Graph-Theoretic Concepts in Computer Science (WG 2015
Claw-free t-perfect graphs can be recognised in polynomial time
A graph is called t-perfect if its stable set polytope is defined by
non-negativity, edge and odd-cycle inequalities. We show that it can be decided
in polynomial time whether a given claw-free graph is t-perfect
Planar Embeddings with Small and Uniform Faces
Motivated by finding planar embeddings that lead to drawings with favorable
aesthetics, we study the problems MINMAXFACE and UNIFORMFACES of embedding a
given biconnected multi-graph such that the largest face is as small as
possible and such that all faces have the same size, respectively.
We prove a complexity dichotomy for MINMAXFACE and show that deciding whether
the maximum is at most is polynomial-time solvable for and
NP-complete for . Further, we give a 6-approximation for minimizing
the maximum face in a planar embedding. For UNIFORMFACES, we show that the
problem is NP-complete for odd and even . Moreover, we
characterize the biconnected planar multi-graphs admitting 3- and 4-uniform
embeddings (in a -uniform embedding all faces have size ) and give an
efficient algorithm for testing the existence of a 6-uniform embedding.Comment: 23 pages, 5 figures, extended version of 'Planar Embeddings with
Small and Uniform Faces' (The 25th International Symposium on Algorithms and
Computation, 2014
Planar 3-SAT with a Clause/Variable Cycle
In the Planar 3-SAT problem, we are given a 3-SAT formula together with its
incidence graph, which is planar, and are asked whether this formula is
satisfiable. Since Lichtenstein's proof that this problem is NP-complete, it
has been used as a starting point for a large number of reductions. In the
course of this research, different restrictions on the incidence graph of the
formula have been devised, for which the problem also remains hard.
In this paper, we investigate the restriction in which we require that the
incidence graph can be augmented by the edges of a Hamiltonian cycle that first
passes through all variables and then through all clauses, in a way that the
resulting graph is still planar. We show that the problem of deciding
satisfiability of a 3-SAT formula remains NP-complete even if the incidence
graph is restricted in that way and the Hamiltonian cycle is given. This
complements previous results demanding cycles only through either the variables
or clauses.
The problem remains hard for monotone formulas, as well as for instances with
exactly three distinct variables per clause. In the course of this
investigation, we show that monotone instances of Planar 3-SAT with exactly
three distinct variables per clause are always satisfiable, thus settling the
question by Darmann, D\"ocker, and Dorn on the complexity of this problem
variant in a surprising way.Comment: Implementing style of DMTCS journa
Recognizing Unit Multiple Intervals Is Hard
Multiple interval graphs are a well-known generalization of interval graphs introduced in the 1970s to deal with situations arising naturally in scheduling and allocation. A d-interval is the union of d intervals on the real line, and a graph is a d-interval graph if it is the intersection graph of d-intervals. In particular, it is a unit d-interval graph if it admits a d-interval representation where every interval has unit length. Whereas it has been known for a long time that recognizing 2-interval graphs and other related classes such as 2-track interval graphs is NP-complete, the complexity of recognizing unit 2-interval graphs remains open. Here, we settle this question by proving that the recognition of unit 2-interval graphs is also NP-complete. Our proof technique uses a completely different approach from the other hardness results of recognizing related classes. Furthermore, we extend the result for unit d-interval graphs for any d ⩾ 2, which does not follow directly in graph recognition problems -as an example, it took almost 20 years to close the gap between d = 2 and d > 2 for the recognition of d-track interval graphs. Our result has several implications, including that recognizing (x, …, x) d-interval graphs and depth r unit 2-interval graphs is NP-complete for every x ⩾ 11 and every r ⩾ 4
Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-Shortest Induced Paths
For vertices and of an -vertex graph , a -trail of is
an induced -path of that is not a shortest -path of . Berger,
Seymour, and Spirkl [Discrete Mathematics 2021] gave the previously only known
polynomial-time algorithm, running in time, to either output a
-trail of or ensure that admits no -trail. We reduce the
complexity to the time required to perform a poly-logarithmic number of
multiplications of Boolean matrices, leading to a largely
improved -time algorithm.Comment: 18 pages, 6 figures, a preliminary version appeared in STACS 202