Reduction Techniques for Graph Isomorphism in the Context of Width Parameters

We study the parameterized complexity of the graph isomorphism problem when parameterized by width parameters related to tree decompositions. We apply the following technique to obtain fixed-parameter tractability for such parameters. We first compute an isomorphism invariant set of potential bags for a decomposition and then apply a restricted version of the Weisfeiler-Lehman algorithm to solve isomorphism. With this we show fixed-parameter tractability for several parameters and provide a unified explanation for various isomorphism results concerned with parameters related to tree decompositions. As a possibly first step towards intractability results for parameterized graph isomorphism we develop an fpt Turing-reduction from strong tree width to the a priori unrelated parameter maximum degree.Comment: 23 pages, 4 figure

A lower bound on opaque sets

It is proved that the total length of any set of countably many rectifiable curves, whose union meets all straight lines that intersect the unit square U, is at least 2.00002. This is the first improvement on the lower bound of 2 by Jones in 1964. A similar bound is proved for all convex sets U other than a triangle. © Akitoshi Kawamura, Sonoko Moriyama, Yota Otachi, and János Pach

On the Recognition of Four-Directional Orthogonal Ray Graphs

Orthogonal ray graphs are the intersection graphs of horizontal and vertical rays (i.e. half-lines) in the plane. If the rays can have any possible orientation (left/right/up/down) then the graph is a 4-directional orthogonal ray graph (4-DORG). Otherwise, if all rays are only pointing into the positive x and y directions, the intersection graph is a 2-DORG. Similarly, for 3-DORGs, the horizontal rays can have any direction but the vertical ones can only have the positive direction. The recognition problem of 2-DORGs, which are a nice subclass of bipartite comparability graphs, is known to be polynomial, while the recognition problems for 3-DORGs and 4-DORGs are open. Recently it has been shown that the recognition of unit grid intersection graphs, a superclass of 4-DORGs, is NP-complete. In this paper we prove that the recognition problem of 4-DORGs is polynomial, given a partition {L,R,U,D} of the vertices of G (which corresponds to the four possible ray directions). For the proof, given the graph G, we first construct two cliques G 1,G 2 with both directed and undirected edges. Then we successively augment these two graphs, constructing eventually a graph TeX with both directed and undirected edges, such that G has a 4-DORG representation if and only if TeX has a transitive orientation respecting its directed edges. As a crucial tool for our analysis we introduce the notion of an S-orientation of a graph, which extends the notion of a transitive orientation. We expect that our proof ideas will be useful also in other situations. Using an independent approach we show that, given a permutation π of the vertices of U (π is the order of y-coordinates of ray endpoints for U), while the partition {L,R} of V ∖ U is not given, we can still efficiently check whether G has a 3-DORG representation

Degree-constrained orientation of maximum satisfaction: graph classes and parameterized complexity

The problem MaxW-Light (MaxW-Heavy) for an undirected graph is to assign a direction to each edge so that the number of vertices of outdegree at most W (resp. at least W) is maximized. It is known that these problems are NP-hard even for fixed W. For example, Max 0-Light is equivalent to the problem of finding a maximum independent set. In this paper, we show that for any fixed constant W, MaxW-Heavy can be solved in linear time for hereditary graph classes for which treewidth is bounded by a function of degeneracy. We show that such graph classes include chordal graphs, circular-arc graphs, d-trapezoid graphs, chordal bipartite graphs, and graphs of bounded clique-width. To have a polynomial-time algorithm for MaxW-Light, we need an additional condition of a polynomial upper bound on the number of potential maximal cliques to apply the metatheorem by Fomin et al. (SIAM J Comput 44:54–87, 2015). The aforementioned graph classes, except bounded clique-width graphs, satisfy such a condition. For graphs of bounded clique-width, we present a dynamic programming approach not using the metatheorem to show that it is actually polynomial-time solvable for this graph class too. We also study the parameterized complexity of the problems and show some tractability and intractability results

A faster parameterized algorithm for pseudoforest deletion

A pseudoforest is a graph where each connected component contains at most one cycle, or alternatively, a graph that can be turned into a forest by removing at most one edge from each connected component. In this paper, we show that the following problem can be solved in O(3knkO(1)) time: given a graph G and an integer k, can we delete at most k vertices from G such that we obtain a pseudoforest? The result improves upon an earlier result by Philip et al. [MFCS 2015] who gave a (nonlinear) 7.56knO(1)-time algorithm both in the exponential factor depending on k as well as in the polynomial factor depending on n

Complexity results for the Spanning Tree Congestion Problem

We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the complexity of this problem. First, we show that for every fixed k and d the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k ≥ 10. For very small values of k however, the problem becomes polynomially solvable. We also show that it is NP-hard to approximate the spanning tree congestion within a factor better than 11/10. On planar graphs, we prove the problem is NP-hard in general, but solvable in linear time for fixed k

Efficient enumeration of maximal k-degenerate induced subgraphs of a chordal graph

In this paper we consider the problem of listing the maximal k-degenerate induced subgraphs of a chordal graph, and propose an output-sensitive algorithm using delay O(m⋅ω(G)) for any n-vertex chordal graph with m edges, where ω(G)≤n is the maximum size of a clique in G. Degeneracy is a well known sparsity measure, and k-degenerate subgraphs are a notion of sparse subgraphs, which generalizes other problems such as independent sets (0-degenerate subgraphs) and forests (1-degenerate subgraphs). Many efficient enumeration algorithms are designed by solving the so-called Extension problem, which asks whether there exists a maximal solution containing a given set of nodes, but no node from a forbidden set. We show that solving this problem is NP-complete for maximal k-degenerate induced subgraphs, motivating the need for additional techniques

Intersection Dimension of Bipartite Graphs

We introduce a concept of intersection dimension of a graphwith respect to a graph class. This generalizes Ferrers dimension, boxicity, and poset dimension, and leads to interesting new problems. We focus in particular on bipartite graph classes defined as intersection graphs of two kinds of geometric objects. We relate well-known graph classes such as interval bigraphs, two-directional orthogonal ray graphs, chain graphs, and (unit) grid intersection graphs with respect to these dimensions. As an application of these graphtheoretic results, we show that the recognition problems for certain graph classes are NP-complete.Theory and Applications of Models of Computation, 11th Annual Conference, TAMC 2014, Chennai, India, April 11-13, 2014. Proceeding