Steiner Trees for Hereditary Graph Classes

Discrete Applied Mathematics, 2015

We design an O(n 3) algorithm to find a minimum weighted coloring of a (P 5 , P 5)-free graph. Furthermore, the same technique can be used to solve the same problem for several classes of graphs, defined by forbidden induced subgraphs, such as (diamond, co-diamond)-free graphs.

Information Processing Letters

A graph is H-free if it does not contain an induced subgraph isomorphic to H. For every integer k and every graph H, we determine the computational complexity of k-Edge Colouring for H-free graphs.

Lecture Notes in Computer Science, 2014

Let Pt and C denote a path on t vertices and a cycle on vertices, respectively. In this paper we study the k-coloring problem for (Pt, C)-free graphs. Maffray and Morel, and Bruce, Hoang and Sawada, have proved that 3-colorability of P5-free graphs has a finite forbidden induced subgraphs characterization, while Hoang, Moore, Recoskie, Sawada, and Vatshelle have shown that k-colorability of P5-free graphs for k ≥ 4 does not. These authors have also shown, aided by a computer search, that 4-colorability of (P5, C5)-free graphs does have a finite forbidden induced subgraph characterization. We prove that for any k, the k-colorability of (P6, C4)-free graphs has a finite forbidden induced subgraph characterization. We provide the full lists of forbidden induced subgraphs for k = 3 and k = 4. As an application, we obtain certifying polynomial time algorithms for 3-coloring and 4-coloring (P6, C4)-free graphs. (Polynomial time algorithms have been previously obtained by Golovach, Paulusma, and Song, but those algorithms are not certifying; in fact they are not efficient in practice, as they depend on multiple use of Ramsey-type results and resulting tree decompositions of very high widths.) To complement these results we show that in most other cases the k-coloring problem for (Pt, C)free graphs is NP-complete. Specifically, for = 5 we show that k-coloring is NP-complete for (Pt, C5)-free graphs when k ≥ 4 and t ≥ 7; for ≥ 6 we show that k-coloring is NP-complete for (Pt, C)-free graphs when k ≥ 5, t ≥ 6; and additionally, for = 7, we show that k-coloring is also NP-complete for (Pt, C7)-free graphs if k = 4 and t ≥ 9. This is the first systematic study of the complexity of the k-coloring problem for (Pt, C)-free graphs. We almost completely classify the complexity for the cases when k ≥ 4, ≥ 4, and identify the last three open cases.

Graphs and Combinatorics, 2019

Indicated coloring is a type of game coloring in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent the realization of this project. The smallest number of colors necessary for Ann to win the game on a graph G (regardless of Ben's strategy) is called the indicated chromatic number of G, denoted by χ i (G). In this paper, we obtain structural characterization of connected {P 5 , K 4 , Kite, Bull}-free graphs which contains an induced C 5 and connected {P 6 , C 5 , K 1,3 }-free graphs that contains an induced C 6 . Also, we prove that {P 5 , K 4 , Kite, Bull}-free graphs that contains an induced C 5 and {P 6 , C 5 , P 5 , K 1,3 }free graphs which contains an induced C 6 are k-indicated colorable for all k ≥ χ(G). In addition, we show that K[C 5 ] is k-indicated colorable for all k ≥ χ(G) and as a consequence, we exhibit that {P 2 ∪ P 3 , C 4 }-free graphs, {P 5 , C 4 }-free graphs are k-indicated colorable for all k ≥ χ(G). This partially answers one of the questions which was raised by A. Grzesik in .

2019

Let H be a fixed undirected graph. A vertex coloring of an undirected input graph G is said to be an \(H\)-Free Coloring if none of the color classes contain H as an induced subgraph. The \(H\)-Free Chromatic Number of G is the minimum number of colors required for an \(H\)-Free Coloring of G. This problem is NP-complete and is expressible in monadic second order logic (MSOL). The MSOL formulation, together with Courcelle’s theorem implies linear time solvability on graphs with bounded tree-width. This approach yields an algorithm with running time \(f(||\varphi ||, t)\cdot n\), where \(||\varphi ||\) is the length of the MSOL formula, t is the tree-width of the graph and n is the number of vertices of the graph. The dependency of \(f(||\varphi ||, t)\) on \(||\varphi ||\) can be as bad as a tower of exponentials.

Arxiv preprint cs/0609083, 2006

Theoretical Computer Science, 2013

In this paper, we consider the selective graph coloring problem. Given an integer k ≥ 1 and a graph G = (V , E) with a partition V 1 , . . . , V p of V , it consists in deciding whether there exists a set V * in G such that |V * ∩ V i | = 1 for all i ∈ {1, . . . , p}, and such that the graph induced by V * is k-colorable. We investigate the complexity status of this problem in various classes of graphs.

arXiv (Cornell University), 2014

We design an O(n 3) algorithm to find a minimum weighted coloring of a (P 5 , P 5)free graph. Furthermore, the same technique can be used to solve the same problem for several classes of graphs, defined by forbidden induced subgraphs, such as (diamond, co-diamond)-free graphs.

Electronic Notes in Discrete Mathematics, 2015

In this paper we will study the complexity of 4-colorability in subclasses of P 6-free graphs. The well known k-colorability problem is NP-complete. It has been shown that if k-colorability is solvable in polynomial time for an induced H-free graph, then every component of H is a path. Recently, Huang [11] has shown several improved complexity results on k-coloring P t-free graphs, where P t is an induced path on t vertices. In summer 2014 only the case k = 4, t = 6 remained open for all k ≥ 4 and all t ≥ 6. Huang conjectures that 4-colorability of P 6-free graphs can be decided in polynomial time. This conjecture has shown to be true for the class of (P 6 , banner)-free graphs by Huang [11] and for the class of (P 6 , C 5)-free graphs by Chudnovsky et al. [6]. In this paper we show that the conjecture also holds for the class of (P 6 , bull, Z 2)-free graphs, for the class of (P 6 , bull, kite)-free graphs, and for the class of (P 6 , chair)-free graphs.

Discrete Mathematics, 2012