Fixed Pattern Graph Research Articles

A Multi-Dimensional Matrix Product—A Natural Tool for Parameterized Graph Algorithms

We introduce the concept of a k-dimensional matrix product D of k matrices A1,…,Ak of sizes n1×n,…,nk×n, respectively, where D[i1,…,ik] is equal to ∑ℓ=1nA1[i1,ℓ]×…×Ak[ik,ℓ]. We provide upper bounds on the time complexity of computing the product and solving related problems of computing witnesses and maximum witnesses of the Boolean version of the product in terms of the time complexity of rectangular matrix multiplication. The multi-dimensional matrix product framework is useful in the design of parameterized graph algorithms. First, we apply our results on the multi-dimensional matrix product to the fundamental problem of detecting/counting copies of a fixed pattern graph in a host graph. The recent progress on this problem has not included complete pattern graphs, i.e., cliques (and their complements, i.e., edge-free pattern graphs, in the induced setting). The fastest algorithms for the aforementioned patterns are based on a reduction to triangle detection/counting. We provide an alternative simple method of detection/counting copies of fixed size cliques based on the multi-dimensional matrix product. It is at least as time efficient as the triangle method in cases of K4 and K5. Next, we show an immediate reduction of the k-dominating set problem to the multi-dimensional matrix product. It implies the W[2] hardness of the problem of computing the k-dimensional Boolean matrix product. Finally, we provide an efficient reduction of the problem of finding the lowest common ancestors for all k-tuples of vertices in a directed acyclic graph to the problem of finding witnesses of the Boolean variant of the multi-dimensional matrix product. Although the time complexities of the algorithms resulting from the aforementioned reductions solely match those of the known algorithms, the advantage of our algorithms is simplicity. Our algorithms also demonstrate the versatility of the multi-dimensional matrix product framework.

Breaking the Hegemony of the Triangle Method in Clique Detection

We consider the fundamental problem of detecting/counting copies of a fixed pattern graph in a host graph. The recent progress on this problem has not included complete pattern graphs, i.e., cliques (and their complements, i.e., edge-free pattern graphs, in the induced setting). The fastest algorithms for the aforementioned patterns are based on a straightforward reduction to triangle detection/counting. We provide an alternative method of detection/counting copies of fixed size cliques based on a multi-dimensional matrix product. It is at least as time efficient as the triangle method in cases of $K_4$ and $K_5.$ The complexity of the multi-dimensional matrix product is of interest in its own rights. We provide also another alternative method for detection/counting $K_r$ copies, again time efficient for $r\in \{ 4, 5 \}$.

Finding Large $H$-Colorable Subgraphs in Hereditary Graph Classes

We study the Max Partial $H$-Coloring problem: given a graph $G$, find the largest induced subgraph of $G$ that admits a homomorphism into $H$, where $H$ is a fixed pattern graph without loops. Not...

Hitting forbidden subgraphs in graphs of bounded treewidth

We study the complexity of a generic hitting problem H-Subgraph Hitting, where given a fixed pattern graph H and an input graph G, the task is to find a set X⊆V(G) of minimum size that hits all subgraphs of G isomorphic to H. In the colorful variant of the problem, each vertex of G is precolored with some color from V(H) and we require to hit only H-subgraphs with matching colors. Standard techniques shows that for every fixed H, the problem is fixed-parameter tractable parameterized by the treewidth of G; however, it is not clear how exactly the running time should depend on treewidth. For the colorful variant, we demonstrate matching upper and lower bounds showing that the dependence of the running time on treewidth of G is tightly governed by μ(H), the maximum size of a minimal vertex separator in H. That is, we show for every fixed H that, on a graph of treewidth t, the colorful problem can be solved in time 2O(tμ(H))⋅|V(G)|, but cannot be solved in time 2o(tμ(H))⋅|V(G)|O(1), assuming the Exponential Time Hypothesis (ETH). Furthermore, we give some preliminary results showing that, in the absence of colors, the parameterized complexity landscape of H-Subgraph Hitting is much richer.

Induced subgraph isomorphism: Are some patterns substantially easier than others?

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. Here, we present two results which, in contrast, provide evidence that no topology of an induced subgraph of fixed size can be substantially easier to detect or count than an independent set of related size.We show that any fixed pattern graph having 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 2k−1 vertices is not easier to detect than an independent set on k vertices, and that an induced cycle on 2k vertices is not easier to detect than an independent set on k vertices. In view of linear time upper bounds on the detection of induced path of length two and three, our lower bound is tight. Similar corollaries hold for the detection of induced complete bipartite graphs and an induced paw and its generalizations.We show also that 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.Next, we show that the so-called diamond and its generalizations on k vertices are not easier to detect as induced subgraphs than an independent set on three vertices or an independent set on k vertices, respectively. For C4, we give a weaker evidence of its hardness in terms of an independent set on three vertices.Finally, we derive several results relating the complexity of the edge-colored variant of induced subgraph isomorphism to that of the standard variant.

Detecting and Counting Small Pattern Graphs

We study the induced subgraph isomorphism problem and the general subgraph isomorphism problem for small pattern graphs. We present a new general method for detecting induced subgraphs of a host graph isomorphic to a fixed pattern graph by reduction to polynomial testing for nonidentity with zero over a field of finite characteristic. It yields new upper time bounds for several pattern graphs on five vertices and provides an alternative combinatorial method for the majority of pattern graphs on four and three vertices. Since our method avoids the large overhead of fast matrix multiplication, it can be of practical interest even for larger pattern graphs. Next, we derive new upper time bounds on counting the number of isomorphisms between a fixed pattern graph with an independent set of size $s$ and a subgraph of the host graph. We also consider a weighted version of the counting problem, when one counts the number of isomorphisms between the pattern graph and lightest subgraphs, providing a slightly slowe...

Efficient Algorithms for Subgraph Listing

Subgraph isomorphism is a fundamental problem in graph theory. In this paper we focus on listing subgraphs isomorphic to a given pattern graph. First, we look at the algorithm due to Chiba and Nishizeki for listing complete subgraphs of fixed size, and show that it cannot be extended to general subgraphs of fixed size. Then, we consider the algorithm due to Ga̧sieniec et al. for finding multiple witnesses of a Boolean matrix product, and use it to design a new output-sensitive algorithm for listing all triangles in a graph. As a corollary, we obtain an output-sensitive algorithm for listing subgraphs and induced subgraphs isomorphic to an arbitrary fixed pattern graph.

Graphs with no 7-wheel subdivision

The topological containment problem, TC(H), has been shown to be polynomial-time solvable for any fixed pattern graph H, but practical algorithms have been developed only for a few specific pattern graphs. Among these are the wheels with four, five, and six spokes. This paper examines the topological containment problem where the pattern graph is a wheel with seven spokes, and gives a result that describes graphs with no W7-subdivision, showing how they can be built up, using certain operations, from smaller ‘pieces’ that meet certain conditions. We also discuss algorithmic aspects of the problem.

The computational complexity of graph contractions II: Two tough polynomially solvable cases

AbstractFor a fixed pattern graph H, let H‐CONTRACTIBILITY denote the problem of deciding whether a given input graph is contractible to H. This article is part II of our study on the computational complexity of the H‐CONTRACTIBILITY problem. In the first article we pinpointed the complexity for all pattern graphs with five vertices except for two pattern graphs H. Here, we present polynomial time algorithms for these two remaining pattern graphs. Interestingly, in all connected cases that are known to be polynomially solvable, the pattern graph H has a dominating vertex, whereas in all cases that are known to be NP‐complete, the pattern graph H does not have a dominating vertex. © 2008 Wiley Periodicals, Inc. NETWORKS, 2008

Structure and Recognition of Graphs with No 6-wheel Subdivision

The subgraph homeomorphism problem has been shown by Robertson and Seymour to be polynomial-time solvable for any fixed pattern graph H. The result, however, is not practical, involving constants that are worse than exponential in |H|. Practical algorithms have only been developed for a few specific pattern graphs, the most recent of these being the wheels with four and five spokes. This paper looks at the subgraph homeomorphism problem where the pattern graph is a wheel with six spokes. The main result is a theorem characterizing graphs that do not contain subdivisions of W 6. We give an efficient algorithm for solving the subgraph homeomorphism problem for W 6. We also give a strengthening of the previous W 5 result.

The computational complexity of graph contractions I: Polynomially solvable and NP‐complete cases

AbstractFor a fixed pattern graph H, let H‐CONTRACTIBILITY denote the problem of deciding whether a given input graph is contractible to H. This paper is part I of our study on the computational complexity of the H‐CONTRACTIBILITY problem. We continue a line of research that was started in 1987 by Brouwer and Veldman, and we determine the computational complexity of the H‐CONTRACTIBILITY problem for certain classes of pattern graphs. In particular, we pinpoint the complexity for all graphs H with five vertices except for two graphs, whose polynomial time algorithms are presented in part II. Interestingly, in all connected cases that are known to be polynomially solvable, the pattern graph H has a dominating vertex, whereas in all cases that are known to be NP‐complete, the pattern graph H does not have a dominating vertex. © 2007 Wiley Periodicals, Inc. NETWORKS, 2008

An approach to the subgraph homeomorphism problem

The subgraph homeomorphism problem for a fixed pattern graph H is stated as follows: given an input graph G = ( V, E), determine whether G has a subgraph homeomorphic to H. We show that the subgraph homeomorphism problem for the fixed graph K 3,3 is solvable in polynomial time, where K 3,3 is the Thomsen graph, one of the Kuratowski graphs used to characterize planar graphs. Specifically, we present an O(| V|)-time algorithm for this problem. This problem was suspected to be NP-complete by Fortune, Hopcroft and Wyllie (1980). We also present several pattern graphs for each of which an O(| V|)-time algorithm exists.