Grundy coloring in some subclasses of bipartite graphs and their complements

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Highlights

  • NP-completeness of Grundy Coloring Decision Problem for Perfect Elimination Bipartite Graphs.

  • NP-completeness of Grundy Coloring Decision Problem for complements of Perfect Elimination Bipartite Graphs.

  • Linear time algorithm for finding an optimal Grundy coloring of chain graphs.

  • Linear time algorithm for finding an optimal Grundy coloring of complements of chain graphs.

  • NP-completeness of Partial Grundy Coloring Decision Problem in complements of Bipartite graphs.

Abstract

A vertex v is a Grundy vertex with respect to a proper k-coloring c of a graph G=(V,E) if v has a neighbor of color j for every j (1j<ik), where i=c(v). A proper k-coloring c of G is called a Grundy k-coloring of G if every vertex is a Grundy vertex with respect to c and the largest integer k such that G admits a Grundy k-coloring is called the Grundy number of G which is denoted as Γ(G). Given a graph G and an integer k, the Grundy number decision problem is to decide whether Γ(G)k. The Grundy number decision problem is known to be NP-complete for bipartite graphs and complement of bipartite graphs. In this paper, we strengthen this result by showing that this problem remains NP-complete for perfect elimination bipartite graphs as well as for complement of perfect elimination bipartite graphs. Further, we give a linear-time algorithm to find the Grundy number of chain graphs, which is a proper subclass of the class of perfect elimination bipartite graphs. We also give a linear-time algorithm to find the Grundy number in complements of chain graphs. A partial Grundy coloring of a graph G is a proper k-coloring of G such that there is at least one Grundy vertex with each color i, 1ik and the partial Grundy number of G, Γ(G), is the largest integer k such that G admits a partial Grundy k-coloring. Given a graph G and an integer k, the partial Grundy number decision problem is to decide whether Γ(G)k. It is known that the partial Grundy number decision problem is NP-complete for bipartite graphs. In this paper, we prove that this problem is NP-complete in the complements of bipartite graphs by showing that the Grundy number and partial Grundy number are equal in complements of bipartite graphs.

Introduction

A proper k-coloring of G=(V,E) is an assignment of k colors to vertices of G such that no two adjacent vertices receive the same color. A k-coloring of a graph G=(V,E) partitions the vertex set V into k independent sets or color classes. A vertex vVi is a Grundy vertex if it is adjacent to at least one vertex in each color class Vj for every j<i. A Grundy k-coloring is a proper k-coloring if every vertex is a Grundy vertex. The Grundy number of G, denoted as Γ(G), is the largest integer k such that there exists a Grundy k-coloring of G. The Grundy number was first studied by Patrick M. Grundy in 1939 in the context of combinatorial games in directed graphs [5]; but was formally introduced later in 1979 by Christen and Selkow [2]. Grundy number has applications in combinatorial games [7], [18]. On the other side, the Grundy number of a graph represents the number of colors used in the worst case coloring that can result from the greedy coloring algorithm. Johnson [11] and McDiarmid [12] also studied how closely Grundy colorings approximate the chromatic number.

A related notion, called partial Grundy coloring, was later introduced in 2003 by P. Erdös et al. [3]. A partial Grundy coloring is a proper k-coloring V=(V1,V2,,Vk) such that there exists at least one Grundy vertex in each color class Vi for all i,1ik. The partial Grundy number of graph G is the maximum integer k such that there exists a partial Grundy coloring of G using k colors and is denoted as Γ(G).

The decision problems associated with Grundy coloring and partial Grundy coloring are as follows:

Grundy Number Decision Problem

Instance: A graph G=(V,E) and a positive integer k.

Question: Does G have a Grundy coloring with at least k colors?

Partial Grundy Number Decision Problem

Instance: A graph G=(V,E) and a positive integer k.

Question: Does G have a partial Grundy coloring with at least k colors?

It is known that the Grundy number decision problem is NP-complete for general graphs [4] and remains NP-complete for bipartite graphs [6], chordal graphs [14] and complement of bipartite graphs [21]. On the other hand, there are polynomial time algorithms to find an optimal Grundy coloring for trees [8] and for partial k-trees [17].

M. Zaker [21] proved the NP-completeness of the Grundy number decision problem in the complement of bipartite graphs by giving a polynomial reduction from the edge domination problem in bipartite graphs, a known NP-complete problem [20]. It implies that if the edge domination problem is NP-complete in class π, then the Grundy number decision problem in the complement of class π is NP-complete by showing a similar reduction, where π is a subclass of the class of bipartite graphs. The Grundy number of complement, G¯, of a bipartite graph G can be computed by the following result of Zaker [22].

Theorem 1 [22]

Let G¯ be the complement of a bipartite graph G. Then Γ(G¯)=nm, where n is the order of G and m is the minimum size of an edge dominating set in G.

A set of edges D of a graph G=(V,E) is said to be an edge dominating set if every edge in ED is adjacent to some edge in D. The edge domination number of G, γ(G) is the cardinality of a minimum edge dominating set of G. The set D of edges is said to be an independent edge dominating set if D is an edge dominating set and no two edges in D are adjacent.

Edge Domination Decision Problem

Instance: A graph G=(V,E) and a positive integer k.

Question: Does G have an edge dominating set with size at most k?

Yannakakis and Gavril [20] have shown that the edge domination decision problem remains NP-complete even for planar graphs as well as for bipartite graphs with maximum degree 3. A. Srinivasan et al. [16] gave an O(nm+n2)-time algorithm to find a minimum edge dominating set of a bipartite permutation graph G, where n and m are the number of vertices and edges, respectively.

The rest of the paper is organized as follows: In Section 2, we give some pertinent definitions and preliminary results. In Section 3, we prove that the Grundy number decision problem remains NP-complete for perfect elimination bipartite graphs and complement of perfect elimination bipartite graphs. In Section 4, we give a linear-time algorithm to find an optimal Grundy coloring of chain graphs. Further, we give a linear-time algorithm to determine the Grundy number of the complement of a chain graph. In Section 5, we show that the partial Grundy number decision problem is NP-complete in the complement of bipartite graphs. Section 6 concludes the paper.

Section snippets

Preliminaries

All the graphs considered in this paper are finite, simple and undirected. For a graph G=(V,E), the set N(v)={uV(G)|uvE} denotes the open neighborhood of a vertex v. A graph G is said to be bipartite if V(G) can be partitioned into two disjoint sets X and Y such that every edge of G joins a vertex in X to another vertex in Y. Such a partition (X,Y) of V is called a bipartition. A bipartite graph with bipartition (X,Y) of V is denoted by G=(X,Y,E).

Let G=(X,Y,E) be a bipartite graph. An edge e=x

NP-completeness results

In this section, we prove that the Grundy number decision problem remains NP-complete even for perfect elimination bipartite graphs, a proper subclass of bipartite graphs. Our proof is inspired by Havet's proof in [6].

Let Kn,n be a complete bipartite graph with partite sets X={x1,x2,, xn} and Y={y1, y2,, yn}. Let Mn,n be the bipartite graph after removing a perfect matching {x1y1,x2y2,,xnyn} from Kn,n. Add a pendant vertex to (n1) vertices of one partite set, say Y, of Mn,n and call it Mn,n

Polynomial-time algorithms

Recall that A. Srinivasan et al. [16] gave an algorithm to find a minimum edge dominating set of a bipartite permutation graph in 1995 with complexity O(nm+n2).

P. Hell and J. Huang [9] have proved that the graph class bipartite permutation graph is equivalent to proper interval bigraphs. They showed that the complement of a proper interval bigraph is a proper circular-arc graph. In view of this and Theorem 1, next corollary follows:

Corollary 2

Grundy coloring problem can be solved for a proper circular-arc

Partial Grundy coloring in complement of bipartite graphs

The partial Grundy number decision problem is NP-complete in (disconnected) chordal graphs and bipartite graphs [15]. Recently, we proved that the problem remains NP-complete in perfect elimination bipartite graphs [13]. We also gave a linear-time algorithm to obtain a partial Grundy coloring with the maximum number of colors in chain graphs. In this section, we prove that the partial Grundy number and Grundy number are the same for the complement of a bipartite graph. Therefore, the results

Conclusion

In this paper, we strengthened the NP-completeness of the Grundy number decision problem by proving that this problem remains NP-complete for perfect elimination bipartite graphs. We also prove that this problem remains NP-complete for complements of perfect elimination bipartite graphs. We give linear-time algorithms to find the Grundy number of chain graphs as well as their complements. It is known that the Grundy coloring problem is polynomial-time solvable in the complement of bipartite

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (22)

  • F. Havet et al.

    The game Grundy number of graphs

    J. Comb. Optim.

    (2013)
  • Cited by (1)

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