Grundy coloring in some subclasses of bipartite graphs and their complements
Introduction
A proper k-coloring of 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 partitions the vertex set V into k independent sets or color classes. A vertex is a Grundy vertex if it is adjacent to at least one vertex in each color class for every . A Grundy k-coloring is a proper k-coloring if every vertex is a Grundy vertex. The Grundy number of G, denoted as , 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 such that there exists at least one Grundy vertex in each color class for all . 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 .
The decision problems associated with Grundy coloring and partial Grundy coloring are as follows:
Grundy Number Decision Problem
Instance: A graph 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 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, , of a bipartite graph G can be computed by the following result of Zaker [22].
Theorem 1 [22] Let be the complement of a bipartite graph G. Then , 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 is said to be an edge dominating set if every edge in is adjacent to some edge in D. The edge domination number of 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 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 -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 , the set denotes the open neighborhood of a vertex v. A graph G is said to be bipartite if 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 of V is called a bipartition. A bipartite graph with bipartition of V is denoted by .
Let be a bipartite graph. An edge
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 be a complete bipartite graph with partite sets , and , , . Let be the bipartite graph after removing a perfect matching from . Add a pendant vertex to vertices of one partite set, say Y, of and call it
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 .
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)
- et al.
Some perfect coloring properties of graphs
J. Comb. Theory, Ser. B
(1979) - et al.
On the equality of the partial Grundy and upper ochromatic numbers of graphs
Discrete Math.
(2003) - et al.
On partial Grundy coloring of bipartite graphs and chordal graphs
Discrete Appl. Math.
(2019) - et al.
An algorithm for partial Grundy number on trees
Discrete Math.
(2005) - et al.
Edge domination on bipartite permutation graphs and cotriangulated graphs
Inf. Process. Lett.
(1995) Results on the Grundy chromatic number of graphs
Discrete Math.
(2006)- et al.
Bipartite Graphs and Their Applications, vol. 131
(1998) - N. Goyal, S. Vishwanathan, Np-completeness of undirected Grundy numbering and related problems, Unpublished manuscript,...
Mathematics and games
Eureka
(1939)- et al.
On the Grundy and b-chromatic numbers of a graph
Algorithmica
(2013)
The game Grundy number of graphs
J. Comb. Optim.
Cited by (1)
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2023, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)