In this paper we consider the Maximum Independent Set problem (MIS) on @math -EPG graphs. EPG (for Edge intersection graphs of Paths on a Grid) was introduced in edgeintersinglebend as the class of graphs whose vertices can be represented as simple paths on a rectangular grid so that two vertices are adjacent if and only if the corresponding paths share at least one edge of the underlying grid. The restricted class @math -EPG denotes EPG-graphs where every path has at most @math bends. The study of MIS on @math -EPG graphs has been initiated in wadsMIS where authors prove that MIS is NP-complete on @math -EPG graphs, and provide a polynomial @math -approximation. In this article we study the approximability and the fixed parameter tractability of MIS on @math -EPG. We show that there is no PTAS for MIS on @math -EPG unless P @math NP, even if there is only one shape of path, and even if each path has its vertical part or its horizontal part of length at most @math . This is optimal, as we show that if all paths have their horizontal part bounded by a constant, then MIS admits a PTAS. Finally, we show that MIS is FPT in the standard parameterization on @math -EPG restricted to only three shapes of path, and @math -hard on @math -EPG. The status for general @math -EPG (with the four shapes) is left open.
In terms of forbidden induced subgraphs, it is also known that @math -EPG graphs exclude induced suns @math with @math , @math , and @math @cite_5 .
@cite_5
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The complexity of the subgraph isomorphism problem where the pattern graph is of fixed size is well known to depend on the topology of the pattern graph. For instance, the larger the maximum independent set of the pattern graph is the more efficient algorithms are known. The situation seems to be substantially different in the case of induced subgraph isomorphism for pattern graphs of fixed size. We present two results which provide evidence that no topology of an induced subgraph of fixed size can be easier to detect or count than an independent set of related size. We show that: Any fixed pattern graph that has a maximum independent set of size k that is disjoint from other maximum independent sets is not easier to detect as an induced subgraph than an independent set of size k. It follows in particular that an induced path on k vertices is not easier to detect than an independent set on ⌈k 2 ⌉ vertices, and that an induced even cycle on k vertices is not easier to detect than an independent set on k 2 vertices. In view of linear time upper bounds on induced paths of length three and four, our lower bound is tight. Similar corollaries hold for the detection of induced complete bipartite graphs and induced complete split graphs. For an arbitrary pattern graph H on k vertices with no isolated vertices, there is a simple subdivision of H, resulting from splitting each edge into a path of length four and attaching a distinct path of length three at each vertex of degree one, that is not easier to detect or count than an independent set on k vertices, respectively. Finally, we show that the so called diamond, paw and C 4 are not easier to detect as induced subgraphs than an independent set on three vertices. Let @math be an induced subgraph of the torus @math . We show that when @math is even and @math divides some power of @math , then for sufficiently large @math the torus @math has a perfect vertex-packing with induced copies of @math . On the other hand, disproving a conjecture of Gruslys, we show that when @math is odd and not a prime power, then there exists @math such that @math divides some power of @math , but there is no @math such that @math has a perfect vertex-packing with copies of @math . We also disprove a conjecture of Gruslys, Leader and Tan by exhibiting a subgraph @math of the @math -dimensional hypercube @math , such that there is no @math for which @math has a perfect edge-packing with copies of @math . We show that there exists a graph G with n spl middot 2 sup O(log* n) nodes, where any forest with n nodes is a node-induced subgraph of G. Furthermore, the result implies the existence of a graph with n sup k 2 sup O(log* n) nodes that contains all n-node graphs of fixed arboricity k as node-induced subgraphs. We provide a lower bound of spl Omega (n sup k ) for the size of such a graph. The upper bound is obtained through a simple labeling scheme for parent queries in rooted trees. The subgraph isomorphism problem involves deciding whether a copy of a pattern graph occurs inside a larger target graph. The non-induced version allows extra edges in the target, whilst the induced version does not. Although both variants are NP-complete, algorithms inspired by constraint programming can operate comfortably on many real-world problem instances with thousands of vertices. However, they cannot handle arbitrary instances of this size. We show how to generate " really hard " random instances for subgraph isomorphism problems, which are computationally challenging with a couple of hundred vertices in the target, and only twenty pattern vertices. For the non-induced version of the problem, these instances lie on a satisfiable unsatisfiable phase transition, whose location we can predict; for the induced variant, much richer behaviour is observed, and constrained-ness gives a better measure of difficulty than does proximity to a phase transition. These results have practical consequences: we explain why the widely researched " filter verify " indexing technique used in graph databases is founded upon a misunderstanding of the empirical hardness of NP-complete problems, and cannot be beneficial when paired with any reasonable subgraph isomorphism algorithm.