Finding a chain graph in a bipartite permutation graph☆

https://doi.org/10.1016/j.ipl.2016.04.006Get rights and content

Highlights

  • •

    Subgraph Isomorphism is studied on graph classes.

  • •

    A polynomial-time algorithm is given for an open case in the literature.

  • •

    Base graphs are bipartite permutation graphs and pattern graphs are chain graphs.

Abstract

We present a polynomial-time algorithm for solving Subgraph Isomorphism where the base graphs are bipartite permutation graphs and the pattern graphs are chain graphs.

Introduction

Given a base graph G and a pattern graph H, Subgraph Isomorphism (SGI for short) asks whether G contains a subgraph isomorphic to H, where a subgraph is a graph obtained by removing some edges and vertices. The problem SGI is in NP and generalizes many NP-complete problems such as Hamiltonicity, Clique, and Bandwidth. Thus SGI is NP-complete in general [2]. The complexity of SGI is studied in many aspects including the parameterized complexity and graph classes. In this paper, we study SGI by restricting input graphs to be in some graph classes. For studies in the parameterized complexity of SGI, see the recent papers by Marx and Pilipczuk [7], Jansen and Marx [4], and the references therein.

Since the problem SGI immediately becomes NP-complete if we allow the input graph class to contain all unions of disjoint paths or all unions of disjoint cliques [1], an easy NP-hardness reduction works for most of graph classes such as forests and cographs. Kijima et al. [5] thus studied a restricted version of SGI that they call Spanning Subgraph Isomorphism (SSGI, for short), where the base and pattern graphs are connected and have the same number of vertices. They showed that SSGI is NP-complete even for bipartite permutation graphs, proper interval graphs, and trivially perfect graphs. On the other hand, they also showed that SGI, the problem without the restrictions, is polynomial-time solvable for chain graphs, cochain graphs, and threshold graphs.

Recently, Konagaya et al. [6] have narrowed the complexity gap by showing that SGI is polynomial-time solvable if the base graphs are proper interval graphs (or the even larger class of chordal graphs) and the pattern graphs are cochain graphs, or if the base graphs are trivially perfect graphs and the pattern graphs are threshold graphs. The complexity of the case where the base graphs are bipartite permutation graphs and the pattern graphs are chain graphs remained unsettled.

In this paper, we study the unsettled case and show that it is polynomial-time solvable. That is, we show that SGI is polynomial-time solvable if the base graphs are bipartite permutation graphs and the pattern graphs are chain graphs.

Section snippets

Preliminaries

Let G=(VG,EG) and H=(VH,EH) be graphs. We say that H is subgraph-isomorphic to G if there exists an injective adjacency-preserving map η from VH to VG; that is, η(u)≠η(v) if u≠v and {η(u),η(v)}∈EG holds for each {u,v}∈EH. We call such a map η a subgraph-isomorphism from H to G. We call G and H the base graph and the pattern graph, respectively. Now SGI can be formally stated as follows:

Problem

SGI

Instance: A pair of graphs G and H.

Question: Is H subgraph-isomorphic to G?

Let G=(V,E) be a graph. For S⊆V

Finding a connected chain graph in a connected 2-layer chain graph

Let G=(X,Y,Z;EG) be a connected 2-layer chain graph with X={x1,…,x|X|}, Y={y1,…,y|Y|}, and Z={z1,…,z|Z|}. Let H=(U,V;EH) be a connected chain graph with U={u1,…,u|U|} and V={v1,…,v|V|}.

Note that since both G and H are connected bipartite graphs, a subgraph-isomorphism η from H to G satisfies either η(U)⊆Y or η(U)⊆X∪Z. Also, η(U)⊆Y implies η(V)⊆X∪Z. In the rest of this section, we assume that we correctly guessed that η(U)⊆Y and η(V)⊆X∪Z since we can perform the algorithm twice.

Concluding remarks

As a final remark, we now show an O(n5) upper bound of the running time, where n is the number of vertices in the base graph. Konagaya et al. [6] showed that by solving O(n) instances of SGI with connected 2-layer chain graphs as the base graphs and connected chain graphs as the pattern graphs, one can solve the problem with bipartite permutation graphs as the base graphs and chain graphs as the pattern graphs. They can produce each instance in linear time. In our algorithm, given a produced

References (8)

There are more references available in the full text version of this article.

Cited by (5)

☆

Partially supported by JSPS/MEXT KAKENHI Grant Numbers 25730003 and 24106004.

View full text
© 2016 Elsevier B.V. All rights reserved.