Subgraph Cover Research Articles

Five‐cycle double cover and shortest cycle cover

AbstractThe 5‐even subgraph cycle double cover conjecture (5‐CDC conjecture) asserts that every bridgeless graph has a 5‐even subgraph double cover. A shortest even subgraph cover of a graph is a family of even subgraphs which cover all the edges of and the sum of their lengths is minimum. It is conjectured that every bridgeless graph has an even subgraph cover with total length at most . In this paper, we study those two conjectures for weak oddness 2 cubic graphs and present a sufficient condition for such graphs to have a 5‐CDC containing a member with many vertices. As a corollary, we show that for every oddness 2 cubic graph satisfying the sufficient condition has a 4‐even subgraph ‐cover with total length at most . We also show that every oddness 2 cubic graph with girth at least 30 has a 5‐CDC containing a member of length at least and thus it has a 4‐even subgraph ‐cover with total length at most .

Approximation Algorithm and FPT Algorithm for Connected-k-Subgraph Cover on Minor-Free Graphs

AbstractGiven a graph G, the minimum Connected-k-Subgraph Cover problem (MinCkSC) is to find a minimum vertex subset C of G such that every connected subgraph of G on k vertices has at least one vertex in C. If furthermore the subgraph of G induced by C is connected, then the problem is denoted as MinCkSC $_{con}$ . In this paper, we first present a PTAS for MinCkSC on an H-minor-free graph, where H is a graph with a constant number of vertices. Then, we design an $O((\omega+1)(2(k-1)(\omega+2))^{3\omega+3})|V|$ -time FPT algorithm for MinCkSC $_{con}$ on a graph with treewidth $\omega$ , based on which we further design an $O(2^{O(\sqrt{t}\log t)}|V|^{O(1)})$ time subexponential FPT algorithm for MinCkSC $_{con}$ on an H-minor-free graph, where t is an upper bound of solution size.

Recognizing simple-triangle graphs by restricted 2-chain subgraph cover

A simple-triangle graph (also known as a PI graph) is the intersection graph of a family of triangles defined by a point on a horizontal line and an interval on another horizontal line. The recognition problem for simple-triangle graphs was a longstanding open problem, and recently a polynomial-time algorithm has been given (Mertzios, 2015). Along with the approach of this paper, we show a simpler recognition algorithm for simple-triangle graphs. To do this, we introduce the following problem: given a bipartite graph G and a set F of edges of G, find a 2-chain subgraph cover of G such that one of two chain subgraphs has no edges in F. We show an O(n2⋅min{m,mˆ})-time algorithm for this problem, where n and m are the number of vertices and edges of G, respectively, and mˆ is the number of non-edges between two independent sets of G. Using this result, we show that the recognition problem for simple-triangle graphs can be solved in O(n2⋅min{m,m̄}) time, where m̄ is the number of non-edges of the given graph.

Subgraph Covers: An Information-Theoretic Approach to Motif Analysis in Networks

Many real world networks contain a statistically surprising number of certain subgraphs, called network motifs. In the prevalent approach to motif analysis, network motifs are detected by comparing subgraph frequencies in the original network with a statistical null model. In this paper we propose an alternative approach to motif analysis where network motifs are defined to be connectivity patterns that occur in a subgraph cover that represents the network using minimal total information. A subgraph cover is defined to be a set of subgraphs such that every edge of the graph is contained in at least one of the subgraphs in the cover. Some recently introduced random graph models that can incorporate significant densities of motifs have natural formulations in terms of subgraph covers and the presented approach can be used to match networks with such models. To prove the practical value of our approach we also present a heuristic for the resulting NP-hard optimization problem and give results for several real world networks.

A Min–Max Property of Chordal Bipartite Graphs with Applications

We show that if G is a bipartite graph with no induced cycles on exactly 6 vertices, then the minimum number of chain subgraphs of G needed to cover E(G) equals the chromatic number of the complement of the square of line graph of G. Using this, we establish that for a chordal bipartite graph G, the minimum number of chain subgraphs of G needed to cover E(G) equals the size of a largest induced matching in G, and also that a minimum chain subgraph cover can be computed in polynomial time. The problems of computing a minimum chain cover and a largest induced matching are NP-hard for general bipartite graphs. Finally, we show that our results can be used to efficiently compute a minimum chain subgraph cover when the input is an interval bigraph.

The induced matching and chain subgraph cover problems for convex bipartite graphs

We present an O ( n 2 ) -time algorithm for computing a maximum cardinality induced matching and a minimum cardinality cover by chain subgraphs for convex bipartite graphs. This improves the previous time bound of O ( m 2 ) .

Induced matchings in asteroidal triple-free graphs

An induced matching M of a graph G is a set of pairwise nonadjacent edges such that no two edges of M are joined by an edge in G. The problem of finding a maximum induced matching is known to be NP-hard even for bipartite graphs of maximum degree four. In this paper, we study the maximum induced matching problem on classes of graphs related to AT-free graphs. We first define a wider class of graphs called the line-asteroidal triple-free (LAT-free) graphs which contains AT-free graphs as a subclass. By examining the square of line graph of LAT-free graphs, we give a characterization of them and apply this for showing that the maximum induced matching problem and a generalization, called the maximum δ-separated matching problem, on LAT-free graphs can be solved in polynomial time. In fact, our result can be extended to the classes of graphs with bounded asteroidal index. Next, we propose a linear-time algorithm for finding a maximum induced matching in a bipartite permutation (bipartite AT-free) graph using the greedy approach. Moreover, we show that using the same technique the minimum chain subgraph cover problem on bipartite permutation graphs can be solved with the same time complexity.

On the complexity of the k-chain subgraph cover problem

The k-chain subgraph cover problem asks if the edge set of a given bipartite graph G is the union of the edge sets of k chain graphs, where each chain graph is a subgraph of G. Although the k-chain subgraph cover problem is known to be NP-complete for the class of bipartite graphs, it is still unknown whether this problem is NP-complete or polynomial-time solvable for subclasses of bipartite graphs. In this paper, we answer this question partially by showing that this problem for an important subclass of bipartite graphs, termed convex bipartite graphs, belongs to not only the class P, but also the class NC. More specifically, we show that the k-chain subgraph cover problem on the convex bipartite graph can be solved in O( m 2) time sequentially or O( log 2 n) time in parallel using O( m 3) processors on the CRCW PRAM, where n and m denote the number of vertices and edges, respectively.

On the 2-Chain Subgraph Cover and Related Problems

In this paper, we provide efficient algorithms for a number of closely related problems. They are: threshold number 2 on split graphs, Ferrers dimension 2, biorder dimension 2, interval dimension 2, and the trapezoid graph recognition problems. We show that all these problems can be reduced to the 2-chain subgraph cover problem in O(n2) time. By introducing a new algorithm to reduce a 2-chain subgraph cover problem to a partial order dimension 2 problem in O(n2) time, we find an O(n2) time algorithm to solve the 2-chain subgraph cover problem. As a consequence, all the previously mentioned problems can be solved in O(n2) time. No other algorithm known to the authors can solve any of these problems in the same time bound.