Complements Of Bipartite Graphs Research Articles

On monophonic position sets in graphs

The general position problem in graph theory asks for the largest set S of vertices of a graph G such that no shortest path of G contains more than two vertices of S. In this paper we consider a variant of the general position problem called the monophonic position problem, obtained by replacing ‘shortest path’ by ‘induced path’. We prove some basic properties and bounds for the monophonic position number of a graph and determine the monophonic position number of some graph families, including unicyclic graphs, complements of bipartite graphs and split graphs. We show that the monophonic position number of triangle-free graphs is bounded above by the independence number. We present realisation results for the general position number, monophonic position number and monophonic hull number. Finally we discuss the complexity of the monophonic position problem.

Grundy coloring in some subclasses of bipartite graphs and their complements

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(1≤j<i≤k), 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, 1≤i≤k 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.

Divisor Graphs that are Complements of Bipartite Graphs

In this paper, we will study some bipartite graphs whose complements are divisor graphs like paths and caterpillars, counterexample for a tree whose complement is not divisor will be presented and powers of some types of trees that have their complements divisor graphs will be classified.

Neighborhood covering and independence on $$P_4$$P4-tidy graphs and tree-cographs

Fil: Duran, Guillermo Alfredo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Calculo; Argentina. Universidad de Chile; Chile. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina

Non-edge orientation and vertex ordering characterizations of some classes of bigraphs

Chordal graphs, permutation graphs, and interval graphs are among many classes of graphs which can be characterized by the existence of certain acyclic orientations and vertex orderings. These types of characterizations exist for some of their bipartite analogues such as chordal bipartite graphs and bipartite permutation graphs. Chvátal proved that a bipartite graph G is chordal bipartite if and only if the complement G¯ of G has a vertex ordering ≺ such that for every induced path abcd in G¯, a≺b implies c≺d. Recently, Le proved that a bipartite graph G is a permutation graph if and only if G¯ admits an acyclic orientation such that for every induced path abcd in G¯, ab is an oriented edge if and only if cd is. Interestingly these orientation and vertex ordering characterizations are stated on the complements of bipartite graphs. We show that interval bigraphs and interval containment bigraphs also admit similar characterizations in terms of vertex orderings and acyclic orientations of their complements.

Scheduling with agreements: new results

In this paper, the problem of scheduling with agreements (SWA) is considered. In scheduling, this consists of a set of jobs non-preemptively on identical machines subject to constraints that only some specific jobs can be scheduled concurrently on different machines. These constraints are given by an agreement graph and the aim is to minimise the makespan. In the case of two machines we extend two NP-hardness results of SWA with processing times at most three that hold for bipartite agreement graphs to more general agreement graphs. Complexity results of SWA are established in the case of split and complement of bipartite graphs. We also present some approximation results for SWA.

On Indicated Coloring of Graphs

Indicated coloring is a graph coloring game in which there are two players collectively coloring 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 $$G$$G, 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 $$G$$G (regardless of Ben's strategy) is called the indicated chromatic number of $$G$$G, and is denoted by $$\chi _i(G)$$?i(G). In this paper, we have shown that cographs, chordal graphs, complement of bipartite graphs, $$\{P_5,K_3\}$${P5,K3}-free graphs and $$\{P_5, \mathrm{paw}\}$${P5,paw}-free graphs are $$k$$k-indicated colorable for all $$k\ge \chi _i(G)$$k??i(G). This provides a partial answer to a question raised in Grzesik (Discret Math 312:3467---3472, 2012). Also we have discussed the Brooks' type result for indicated coloring.

On minimal forbidden subgraph characterizations of balanced graphs

A graph is balanced if its clique-matrix contains no edge–vertex incidence matrix of an odd chordless cycle as a submatrix. While a forbidden induced subgraph characterization of balanced graphs is known, there is no such characterization by minimal forbidden induced subgraphs. In this work, we provide minimal forbidden induced subgraph characterizations of balanced graphs restricted to graphs that belong to one of the following graph classes: complements of bipartite graphs, line graphs of multigraphs, and complements of line graphs of multigraphs. These characterizations lead to linear-time recognition algorithms for balanced graphs within the same three graph classes.

Krausz dimension and its generalizations in special graph classes

Graph Theory A Krausz (k,m)-partition of a graph G is a decomposition of G into cliques, such that any vertex belongs to at most k cliques and any two cliques have at most m vertices in common. The m-Krausz dimension kdimm(G) of the graph G is the minimum number k such that G has a Krausz (k,m)-partition. In particular, 1-Krausz dimension or simply Krausz dimension kdim(G) is a well-known graph-theoretical parameter. In this paper we prove that the problem "kdim(G)≤3" is polynomially solvable for chordal graphs, thus partially solving the open problem of P. Hlineny and J. Kratochvil. We solve another open problem of P. Hlineny and J. Kratochvil by proving that the problem of finding Krausz dimension is NP-hard for split graphs and complements of bipartite graphs. We show that the problem of finding m-Krausz dimension is NP-hard for every m≥1, but the problem "kdimm(G)≤k" is is fixed-parameter tractable when parameterized by k and m for (∞,1)-polar graphs. Moreover, the class of (∞,1)-polar graphs with kdimm(G)≤k is characterized by a finite list of forbidden induced subgraphs for every k,m≥1.

Chromatic Polynomials of Complements of Bipartite Graphs

We define a biclique to be the complement of a bipartite graph, consisting of two cliques joined by a number of edges. In this paper we study algebraic aspects of the chromatic polynomials of these graphs. We derive a formula for the chromatic polynomial of an arbitrary biclique, and use this to give certain conditions under which two of the graphs have chromatic polynomials with the same splitting field. Finally, we use a subfamily of bicliques to prove the cubic case of the α + n conjecture, by showing that for any cubic integer α, there is a natural number n such that α + n is a chromatic root.

Formula omitted]-coloring of tight graphs

A coloring c of a graph G=(V,E) is a b-coloring if in every color class there is a vertex whose neighborhood intersects every other color class. The b-chromatic number of G, denoted χb(G), is the greatest integer k such that G admits a b-coloring with k colors. A graph G is tight if it has exactly m(G) vertices of degree m(G)−1, where m(G) is the largest integer m such that G has at least m vertices of degree at least m−1. Determining the b-chromatic number of a tight graph G is NP-hard even for a connected bipartite graph Kratochvil et al. (2002) [18]. In this paper we show that it is also NP-hard for a tight chordal graph. We also show that the b-chromatic number of a split graph can be computed in polynomial time. We then define the b-closure and the partial b-closure of a tight graph, and use these concepts to give a characterization of tight graphs whose b-chromatic number is equal to m(G). This characterization is used to propose polynomial-time algorithms for deciding whether χb(G)=m(G) for tight graphs that are complement of bipartite graphs, P4-sparse and block graphs. We also generalize the concept of pivoted tree introduced by Irving and Manlove (1999) [13] and show its relation with the b-chromatic number of tight graphs.

On Polar, Trivially Perfect Graphs

&lt;p&gt;During the last decades, different types of decompositions have been processed in the field of graph theory. In various problems, for example in the construction of recognition algorithms, frequently appears the so-called weakly decomposition of graphs.&lt;br /&gt;Polar graphs are a natural extension of some classes of graphs like bipartite graphs, split graphs and complements of bipartite graphs. Recognizing a polar graph is known to be NP-complete. For this class of graphs, polynomial algorithms for the maximum stable set problem are unknown and algorithms for the dominating set problem are also NP-complete.&lt;br /&gt;In this paper we characterize the polar graphs using the weakly decomposition, give a polynomial time algorithm for recognizing graphs that are both trivially perfect and polar, and directly calculate the domination number. For the stability number and clique number, we give polynomial time algorithms. &lt;/p&gt;

Locally connected spanning trees in cographs, complements of bipartite graphs and doubly chordal graphs

A spanning tree T of a graph G = ( V , E ) is called a locally connected spanning tree if the set of all neighbors of v in T induces a connected subgraph of G for all v ∈ V . The problem of recognizing whether a graph admits a locally connected spanning tree is known to be NP-complete even when the input graphs are restricted to chordal graphs. In this paper, we propose linear time algorithms for finding locally connected spanning trees in cographs, complements of bipartite graphs and doubly chordal graphs, respectively.

Game-perfect graphs

A graph coloring game introduced by Bodlaender (Int J Found Comput Sci 2:133–147, 1991) as coloring construction game is the following. Two players, Alice and Bob, alternately color vertices of a given graph G with a color from a given color set C, so that adjacent vertices receive distinct colors. Alice has the first move. The game ends if no move is possible any more. Alice wins if every vertex of G is colored at the end, otherwise Bob wins. We consider two variants of Bodlaender’s graph coloring game: one (A) in which Alice has the right to have the first move and to miss a turn, the other (B) in which Bob has these rights. These games define the A-game chromatic number resp. the B-game chromatic number of a graph. For such a variant g, a graph G is g-perfect if, for every induced subgraph H of G, the clique number of H equals the g-game chromatic number of H. We determine those graphs for which the game chromatic numbers are 2 and prove that the triangle-free B-perfect graphs are exactly the forests of stars, and the triangle-free A-perfect graphs are exactly the graphs each component of which is a complete bipartite graph or a complete bipartite graph minus one edge or a singleton. From these results we may easily derive the set of triangle-free game-perfect graphs with respect to Bodlaender’s original game. We also determine the B-perfect graphs with clique number 3. As a general result we prove that complements of bipartite graphs are A-perfect.

Polar cographs

Polar graphs are a natural extension of some classes of graphs like bipartite graphs, split graphs and complements of bipartite graphs. A graph is ( s , k ) -polar if there exists a partition A , B of its vertex set such that A induces a complete s-partite graph (i.e., a collection of at most s disjoint stable sets with complete links between all sets) and B a disjoint union of at most k cliques (i.e., the complement of a complete k-partite graph). Recognizing a polar graph is known to be NP-complete. These graphs have not been extensively studied and no good characterization is known. Here we consider the class of polar graphs which are also cographs (graphs without induced path on four vertices). We provide a characterization in terms of forbidden subgraphs. Besides, we give an algorithm in time O ( n ) for finding a largest induced polar subgraph in cographs; this also serves as a polar cograph recognition algorithm. We examine also the monopolar cographs which are the ( s , k ) -polar cographs where min ( s , k ) ⩽ 1 . A characterization of these graphs by forbidden subgraphs is given. Some open questions related to polarity are discussed.

On the probabilistic minimum coloring and minimum [formula omitted]-coloring

We study a robustness model for the minimum coloring problem, where any vertex v i of the input-graph G ( V , E ) has some presence probability p i . We show that, under this model, the original coloring problem gives rise to a new coloring version (called Probabilistic Min Coloring) where the objective becomes to determine a partition of V into independent sets S 1 , S 2 , … , S k , that minimizes the quantity ∑ i = 1 k f ( S i ) , where, for any independent set S i , i = 1 , … , k , f ( S i ) = 1 - ∏ v j ∈ S i ( 1 - p j ) . We show that Probabilistic Min Coloring is NP-hard and design a polynomial time approximation algorithm achieving non-trivial approximation ratio. We then focus ourselves on probabilistic coloring of bipartite graphs and show that the problem of determining the best k-coloring (called Probabilistic Min k -Coloring) is NP-hard, for any k ⩾ 3 . We finally study Probabilistic Min Coloring and Probabilistic Min k -Coloring in a particular family of bipartite graphs that plays a crucial role in the proof of the NP-hardness result just mentioned, and in complements of bipartite graphs.

On the Maximum Weight Clique Problem

We introduce several new classes of graphs on which the maximum-weight clique problem is solvable in polynomial time. Their common feature, and the central idea of our algorithms, is that every clique of any of our graphs is contained in some member of a polynomial-sized collection of induced subgraphs that are complements of bipartite graphs.