The Balanced Connected Subgraph Problem

European Journal of Operational Research, 2015

In this paper we present the Selective Graph Coloring Problem, a generalization of the standard graph coloring problem as well as several of its possible applications. Given a graph with a partition of its vertex set into several clusters, we want to select one vertex per cluster such that the chromatic number of the subgraph induced by the selected vertices is minimum. This problem appeared in the literature under different names for specific models and its complexity has recently been studied for different classes of graphs. Here, we describe different models -some already discussed in previous papers and some new ones -in very different contexts under a unified framework based on this graph problem. We point out similarities between these models, offering a new approach to solve them, and show some generic situations where the selective graph coloring problem may be used. We focus on specific graph classes motivated by each model, and we briefly discuss the complexity of the selective graph coloring problem in each one of these graph classes and point out interesting future research directions.

Lecture Notes in Computer Science, 2013

We consider graphs without loops or parallel edges in which every edge is assigned + or −. Such a signed graph is balanced if its vertex set can be partitioned into parts V1 and V2 such that all edges between vertices in the same part have sign + and all edges between vertices of different parts have sign − (one of the parts may be empty). It is well-known that every connected signed graph with n vertices and m edges has a balanced subgraph with at least m 2 + n−1 4 edges and this bound is tight. We consider the following parameterized problem: given a connected signed graph G with n vertices and m edges, decide whether G has a balanced subgraph with at least m 2 + n−1 4 + k 4 edges, where k is the parameter. We obtain an algorithm for the problem of runtime 8 k (kn) O(1). We also prove that for each instance (G, k) of the problem, in polynomial time, we can either solve (G, k) or produce an equivalent instance (G ′ , k ′) such that k ′ ≤ k and |V (G ′)| = O(k 3). Our first result generalizes a result of Crowston, Jones and Mnich (ICALP 2012) on the corresponding parameterization of Max Cut (when every edge of G has sign −). Our second result generalizes and significantly improves the corresponding result of Crowston, Jones and Mnich for MaxCut: they showed that |V (G ′)| = O(k 5).

Information Processing Letters, 1981

Mathematica Bohemica, 2016

We consider, for a positive integer k, induced subgraphs in which each component has order at most k. Such a subgraph is said to be k-divided. We show that finding large induced subgraphs with this property is NP-complete. We also consider a related graph-coloring problem: how many colors are required in a vertex coloring in which each color class induces a k-divided subgraph. We show that the problem of determining whether some given number of colors suffice is NP-complete, even for 2-coloring a planar trianglefree graph. Lastly, we consider Ramsey-type problems where graphs or their complements with large enough order must contain a large k-divided subgraph. We study the asymptotic behavior of "k-divided Ramsey numbers". We conclude by mentioning a number of open problems.

Discrete Applied Mathematics, 2009

We study the complexity of the problem of deciding the existence of a spanning subgraph of a given graph, and of that of finding a maximum (weight) such subgraph. We establish some general relations between these problems, and we use these relations to obtain new NPcompleteness results for maximum (weight) spanning subgraph problems from analogous results for existence problems and from results in extremal graph theory. On the positive side, we provide a decomposition method for the maximum (weight) spanning chordal subgraph problem that can be used, e.g., to obtain a linear (or O(n log n)) time algorithm for such problems in graphs with vertex degree bounded by 3.

HAL (Le Centre pour la Communication Scientifique Directe), 2023

Given an undirected graph G and a real edge-weight vector, the connected subgraph problem consists of finding a maximum-weight subset of edges which induces a connected subgraph of G. In this paper, we establish a link between the complexity of the connected subgraph problem and the matching number. We study the separation problem associated with the Matching-partition inequalities wich are introduced by Didi Biha et al. [4] for the connected subgraph polytope.

Theoretical Computer Science, 2017

An edge-bicolored graph G = (V , R ∪ B) is called Red/Blue-split graph if there exists a partition I B and I R of V such that I B and I R are independent sets in (V , B) and (V , R), respectively. Red/Blue-split graphs generalize several well studied graph classes including split graphs, bipartite graphs and König-Egerváry graphs. In this paper we consider the algorithmic complexity of various optimization problems like minimum edge (or vertex) deletion and maximum edge (or vertex) induced subgraph related to Red/Blue split graphs. All these problems are N P-hard and thus we look at them from algorithmic paradigms that are meant for coping with N P-hardness. We obtain various hardness as well as algorithmic results for these problems in the realm of approximation algorithms and parameterized complexity. The main tool we use to obtain all our results is polynomial time transformations between appropriate problems. On the way, we also resolve some problems related to inapproximability about certain optimization problems mentioned by Korach et al. (2006) [17].

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.

Lecture Notes in Computer Science

A graph H is p-edge colorable if there is a coloring ψ : E(H) → {1, 2,. .. , p}, such that for distinct uv, vw ∈ E(H), we have ψ(uv) = ψ(vw). The Maximum Edge-Colorable Subgraph problem takes as input a graph G and integers l and p, and the objective is to find a subgraph H of G and a p-edge-coloring of H, such that |E(H)| ≥ l. We study the above problem from the viewpoint of Parameterized Complexity. We obtain FPT algorithms when parameterized by: (1) the vertex cover number of G, by using Integer Linear Programming, and (2) l, a randomized algorithm via a reduction to Rainbow Matching, and a deterministic algorithm by using color coding, and divide and color. With respect to the parameters p + k, where k is one of the following: (1) the solution size, l, (2) the vertex cover number of G, and (3) l − mm(G), where mm(G) is the size of a maximum matching in G; we show that the (decision version of the) problem admits a kernel with O(k • p) vertices. Furthermore, we show that there is no kernel of size O(k 1− • f (p)), for any > 0 and computable function f , unless NP ⊆ coNP/poly.

arXiv (Cornell University), 2009

In a bounded max-coloring of a vertex/edge weighted graph, each color class is of cardinality at most b and of weight equal to the weight of the heaviest vertex/edge in this class. The bounded max-vertex/edge-coloring problems ask for such a coloring minimizing the sum of all color classes' weights. In this paper we present complexity results and approximation algorithms for those problems on general graphs, bipartite graphs and trees. We first show that both problems are polynomial for trees, when the number of colors is fixed, and H b approximable for general graphs, when the bound b is fixed. For the bounded max-vertex-coloring problem, we show a 17/11-approximation algorithm for bipartite graphs, a PTAS for trees as well as for bipartite graphs when b is fixed. For unit weights, we show that the known 4/3 lower bound for bipartite graphs is tight by providing a simple 4/3 approximation algorithm. For the bounded max-edge-coloring problem, we prove approximation factors of 3 − 2/ √ 2b, for general graphs, min{e, 3 − 2/ √ b}, for bipartite graphs, and 2, for trees. Furthermore, we show that this problem is NP-complete even for trees. This is the first complexity result for max-coloring problems on trees.

Discrete Applied Mathematics, 1998

It is well-known that the GRAPH 3.COLORABILITY problem, deciding whether a given graph has a stable set whose deletion results in a bipartite graph, is NP-complete. We prove the following related theorems: It is NP-complete to decide whether a graph has a stable set whose deletion results in (1) a tree or (2) a trivially perfect graph, and there is a polynomial algorithm to decide if a given graph has a stable set whose deletion results in (3) the complement of a bipartite graph, (4) a split graph or (5) a threshold graph. 0 1998 Elsevier Science B.V. All rights reserved.

Discrete Mathematics, 2009

We offer the following structural result: every triangle-free graph G of maximum degree 3 has 3 matchings which collectively cover at least 1 − 2 3 γo(G) of its edges, where γ o (G) denotes the odd girth of G. In particular, every triangle-free graph G of maximum degree 3 has 3 matchings which cover at least 13/15 of its edges. The Petersen graph, where we can 3-edge-color at most 13 of its 15 edges, shows this to be tight. We can also cover at least 6/7 of the edges of any simple graph of maximum degree 3 by means of 3 matchings; again a tight bound.

Journal of Discrete Algorithms, 2011

In the Connected Red-Blue Dominating Set problem we are given a graph G whose vertex set is partitioned into two parts R and B (red and blue vertices), and we are asked to find a connected subgraph induced by a subset S of B such that each red vertex of G is adjacent to some vertex in S. The problem can be solved in O * (2 n−|B| ) time by reduction to the Weighted Steiner Tree problem. Combining exhaustive enumeration when |B| is small with the Weighted Steiner Tree approach when |B| is large, solves the problem in O * (1.4143 n ). In this paper we present a first non-trivial exact algorithm whose running time is in O * (1.3645 n ). We use our algorithm to solve the Connected Dominating Set problem in O * . This improves the current best known algorithm, which used sophisticated run-time analysis via the measure and conquer technique to solve the problem in O * (1.8966 n ).

Discrete Mathematics, 2003

We consider the subchromatic number χS(G) of graph G, which is the minimum order of all partitions of V(G) with the property that each class in the partition induces a disjoint union of cliques. Here we establish several bounds on subchromatic number. For example, we consider the maximum subchromatic number of all graphs of order n and in so doing answer a question posed in Jensen and Toft (Graph Coloring Problems, Wiley, New York, 1995). We also consider bounds on χS(G) when the size and genus of G are known. We also consider the parameter when applied to planar and outerplanar graphs. It is known that the problem of determining whether χS(G)⩽k is NP-complete for all k⩾2. We extend this by showing it is NP-complete for k=2 even when restricted to the class of planar triangle-free graphs with maximum degree four. As a corollary, we see that showing a planar triangle-free graph of maximum degree four has a 1-defective chromatic number of two is NP-complete, answering a question of Cowen et al. (J. Graph Theory 24(3) (1997) 205–219). We show that determining whether χS(G)⩽3 is NP-complete for planar graphs. We consider the subchromatic number of cartesian products of complete graphs and show a correspondence with a natural covering of matrices. We close by producing bounds on the subchromatic number in terms of chromatic number as well as the product of clique number with chromatic number. Sharpness for graphs with fixed clique size is discussed.

Information Processing Letters, 2002

A partition of the vertices of a graph G into k pairwise disjoint sets V 1 , . . . , V k is called an (r 1 , . . . , r k )-subcoloring if the subgraph G i of G induced by V i , 1 i k, consists of disjoint complete subgraphs, each of which has cardinality no more than r i . Due to Erdős and Albertson et al., independently, every cubic (i.e., 3-regular) graph has a (2, 2)-subcoloring. Albertson et al. then asked for cubic graphs having (1, 2)-subcolorings. We point out in this paper that this question is algorithmically difficult by showing that recognizing (1, 2)-subcolorable cubic graphs is NP-complete, even when restricted to triangle-free planar graphs. 