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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. 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 2 k - 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 C 4 , 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.

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...

Given a pattern graph H with l edges, and a host graph G guaranteed to contain at most one occurrence of a subgraph isomorphic to H , we show that the time complexity of the problem of finding such an occurrence if any in G as well as that of the decision version of the problem are within a multiplicative factor O l of the time complexity for the corresponding problem in the general case, when G may contain several occurrences of H . It follows that for pattern graphs of constant size, the aforementioned uniqueness guarantee cannot yield any asymptotic speed up. We also derive analogous results with the analogous multiplicative factor linear in the number of vertices of H in the induced case when occurrences of induced subgraphs of G isomorphic to H are sought.

We present a general technique for detecting and counting small subgraphs. It consists of forming special linear combinations of the numbers of occurrences of different induced subgraphs of fixed size in a graph. These combinations can be efficiently computed by rectangular matrix multiplication. Our two main results utilizing the technique are as follows. Let $H$ be a fixed graph with $k$ vertices and an independent set of size $s.$ 1. Detecting if an $n$-vertex graph contains a (not necessarily induced) subgraph isomorphic to $H$ can be done in time $O(n^{\omega(\lceil (k-s)/2 \rceil, 1, \lfloor (k-s)/2 \rfloor )})$, where $\omega (p,q,r)$ is the exponent of fast arithmetic matrix multiplication of an $n^p\times n^q$ matrix by an $n^q\times n^r$ matrix. 2. When $s=2,$ counting the number of (not necessarily induced) subgraphs isomorphic to $H$ can be done in the same time, i.e., in time $O(n^{\omega(\lceil (k-2)/2 \rceil, 1, \lfloor (k-2)/2 \rfloor )}).$ It follows in particular that we can count the nu...

We describe a simple combinatorial approximation algorithm for finding a shortest (simple) cycle in an undirected graph. Given an adjacency-list representation of an undirected graph G with n vertices and unknown girth k, our algorithm returns with high probability a cycle of length at most 2k for even k and 2k+2 for odd k, in time O(n^3^2logn). Thus, in general, it yields a 223 approximation. For a weighted, undirected graph, with non-negative edge weights in the range {1,2,...,M}, we present a simple combinatorial 2-approximation algorithm for a minimum weight (simple) cycle that runs in time O(n^2logn(logn+logM)).

For a set of rooted, unordered, distinctly leaf-labeled trees, the NP-hard maximum agreement subtree problem (MAST) asks for a tree contained (up to isomorphism or homeomorphism) in all of the input trees with as many labeled leaves as possible. We study the ordered variants of MAST where the trees are uniformly or non-uniformly ordered. We provide the first known polynomial-time algorithms for the uniformly and non-uniformly ordered homeomorphic variants as well as the uniformly and non-uniformly ordered isomorphic variants of MAST. Our algorithms run in time $O(kn^3)$, $O(n^3 \min \{kn,\, n+\log^{k-1}n\})$, $O(kn^3)$, and $O(n^3 \min \{kn,\, n+\log^{k-1}n\})$, respectively, where n is the number of leaf labels and k is the number of input trees.

We consider the following clustering problems: given an undirected graph, partition its vertices into disjoint clusters such that each cluster forms a clique and the number of edges within the clusters is maximized (Max-ECP), or the number of edges between clusters is minimized (Min-ECP). These problems arise naturally in the DNA clone classification. We investigate the hardness of finding such partitions and provide approximation algorithms. Further, we show that greedy strategies yield constant factor approximations for graph classes for which maximum cliques can be found efficiently.

We study the induced subgraph isomorphism problem and the general subgraph isomorphism problem for small pattern graphs.

Given a pattern graph H of fixed size, and a host graph G guaranteed to contain at most one occurrence of a subgraph isomorphic to H, we show that both the problem of finding such an occurrence (if any) as well as the decision version of the problem are as hard as in the general case when G may contain several occurrences of H.

We describe a simple combinatorial approximation algorithm for finding a shortest (simple) cycle in an undirected graph. For an undirected graph G of unknown girth k, our algorithm returns with high probability a cycle of length at most 2k for even k and 2k + 2 for odd k, in time O(n3/2 √log n). Thus, in general, it yields a 2 2/3 approximation. We study also the problem of finding a simple cycle of minimum total weight in an undirected graph with nonnegative edge weights. We present a simple combinatorial 2-approximation algorithm for a minimum weight (simple) cycle in an undirected graph with nonnegative integer edge weights in the range {1, 2, ..., M}. This algorithm runs in time O(n2 log n log M).