Co-bipartite Graphs Research Articles

Hadwiger Number of Graphs with Small Chordality

The Hadwiger number of a graph $G$ is the largest integer $h$ such that $G$ has the complete graph $K_h$ as a minor. We show that the problem of determining the Hadwiger number of a graph is \sf NP-hard on co-bipartite graphs but can be solved in polynomial time on cographs and on bipartite permutation graphs. We also consider a natural generalization of this problem that asks for the largest integer $h$ such that $G$ has a minor with $h$ vertices and diameter at most $s$. We show that this problem can be solved in polynomial time on AT-free graphs when $s\geq 2$ but is \sf NP-hard on chordal graphs for every fixed $s\geq 2$.

Efficiently decomposing, recognizing and triangulating hole-free graphs without diamonds

A graph is hole- and diamond-free (HD-free) if none of its induced subgraphs is isomorphic to a chordless cycle of length at least five or to a diamond. Using the clique separator approach and the simple structure of atoms of HD-free graphs, we show how to recognize HD-free graphs in time O(n2). One of the main tools is Lexicographic Breadth-First Search (LexBFS); we give two new properties of LexBFS which are essential for reaching the time bound and which hold for any graph. Moreover, we find minimal triangulations of HD-free graphs in time O(n2), introducing efficient algorithms for the minimal triangulation of matched co-bipartite graphs and chordal bipartite graphs.

On the number of minimal dominating sets on some graph classes

A dominating set in a graph is a subset of vertices such that each vertex is either in the dominating set or adjacent to some vertex in the dominating set. It is known that graphs have at most O(1.7159n) minimal dominating sets. In this paper, we establish upper bounds on this maximum number of minimal dominating sets for split graphs, cobipartite graphs and interval graphs. For each of these graph classes, we provide an algorithm to enumerate them. For split and interval graphs, we show that the number of minimal dominating sets is at most 3n/3≈1.4423n, which is the best possible bound. This settles a conjecture by Couturier et al. (SOFSEM 2012, [1]). For cobipartite graphs, we lower the O(1.5875n) upper bound from Couturier et al. to O(1.4511n).

$$b$$ b -Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree-Cographs

A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by \chi_b(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of G, and every induced subgraph H_2 of H_1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: - We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. - We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at most a given threshold. - We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. - Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic.

Block decomposition approach to compute a minimum geodetic set

In this paper, we develop a divide-and-conquer approach, called block decomposition, to solve the minimum geodetic set problem. This provides us with a unified approach for all graphs admitting blocks for which the problem of finding a minimum geodetic set containing a given set of vertices (g -extension problem) can be efficiently solved. Our method allows us to derive linear time algorithms for the minimum geodetic set problem in (a proper superclass of) block-cacti and monopolar chordal graphs. Also, we show that hull sets and geodetic sets of block-cacti are the same, and the minimum geodetic set problem is NP-hard in cobipartite graphs. We conclude by pointing out several interesting research directions.

The (Weighted) Metric Dimension of Graphs: Hard and Easy Cases

Given an input undirected graph $$G=(V,E)$$G=(V,E), we say that a vertex $$\ell $$l separates $$u$$u from $$v$$v (where $$u,v\in V$$u,v?V) if the distance between $$u$$u and $$\ell $$l differs from the distance from $$v$$v to $$\ell $$l. A set of vertices $$L\subseteq V$$L⊆V is a feasible solution if for every pair of vertices, $$u,v\in V$$u,v?V ($$u\ne v$$u?v), there is a vertex $$\ell \in L$$l?L that separates $$u$$u from $$v$$v. Such a feasible solution is called a landmark set, and the metric dimension of a graph is the minimum cardinality of a landmark set. Here, we extend this well-studied problem to the case where each vertex $$v$$v has a non-negative cost, and the goal is to find a feasible solution with a minimum total cost. This weighted version is NP-hard since the unweighted variant is known to be NP-hard. We show polynomial time algorithms for the cases where $$G$$G is a path, a tree, a cycle, a cograph, a $$k$$k-edge-augmented tree (that is, a tree with additional $$k$$k edges) for a constant value of $$k$$k, and a (not necessarily complete) wheel. The results for paths, trees, cycles, and complete wheels extend known polynomial time algorithms for the unweighted version, whereas the other results are the first known polynomial time algorithms for these classes of graphs even for the unweighted version. Next, we extend the set of graph classes for which computing the unweighted metric dimension of a graph is known to be NP-hard. We show that for split graphs, bipartite graphs, co-bipartite graphs, and line graphs of bipartite graphs, the problem of computing the unweighted metric dimension of the graph is NP-hard.

Competition Numbers, Quasi-line Graphs, and Holes

The competition graph of an acyclic directed graph $D$ is the undirected graph on the same vertex set as $D$ in which two distinct vertices are adjacent if they have a common out-neighbor in $D$. The competition number of an undirected graph $G$ is the least number of isolated vertices that have to be added to $G$ to make it the competition graph of an acyclic directed graph. We resolve two conjectures concerning competition graphs. First, we prove a conjecture of Opsut by showing that the competition number of every quasi-line graph is at most 2. Recall that a quasi-line graph, also called a locally co-bipartite graph, is a graph for which the neighborhood of every vertex can be partitioned into at most two cliques. To prove this conjecture we devise an alternative characterization of quasi-line graphs to the one by Chudnovsky and Seymour. Second, we prove a conjecture of Kim by showing that the competition number of any graph is at most one greater than the number of holes in the graph. Our methods also allow us to prove a strengthened form of this conjecture recently proposed by Kim et al., showing that the competition number of any graph is at most one greater than the dimension of the subspace of the cycle space spanned by the holes.

Matrix partitions of split graphs

Matrix partition problems generalize a number of natural graph partition problems, and have been studied for several standard graph classes. We prove that each matrix partition problem has only finitely many minimal obstructions for split graphs. Previously such a result was only known for the class of cographs. (In particular, there are matrix partition problems which have infinitely many minimal chordal obstructions.) We provide (close) upper and lower bounds on the maximum size of a minimal split obstruction. This shows for the first time that some matrices have exponential-sized minimal obstructions of any kind (not necessarily split graphs). We also discuss matrix partitions for bipartite and co-bipartite graphs.

Fully Dynamic Recognition of Proper Circular-Arc Graphs

We present a fully dynamic algorithm for the recognition of proper circular-arc (PCA) graphs. The allowed operations on the graph involve the insertion and removal of vertices (together with its incident edges) or edges. Edge operations cost O(log n) time, where n is the number of vertices of the graph, while vertex operations cost O(log n + d) time, where d is the degree of the modified vertex. We also show incremental and decremental algorithms that work in O(1) time per inserted or removed edge. As part of our algorithm, fully dynamic connectivity and co-connectivity algorithms that work in O(log n) time per operation are obtained. Also, an O(\Delta) time algorithm for determining if a PCA representation corresponds to a co-bipartite graph is provided, where \Delta\ is the maximum among the degrees of the vertices. When the graph is co-bipartite, a co-bipartition of each of its co-components is obtained within the same amount of time.

On opposition graphs, coalition graphs, and bipartite permutation graphs

A graph is an opposition graph, respectively, a coalition graph, if it admits an acyclic orientation which puts the two end-edges of every chordless 4-vertex path in opposition, respectively, in the same direction. Opposition and coalition graphs have been introduced and investigated in connection to perfect graphs. Recognizing and characterizing opposition and coalition graphs still remain long-standing open problems. The present paper gives characterizations for co-bipartite opposition graphs and co-bipartite coalition graphs, and for bipartite opposition graphs. Implicit in our argument is a linear time recognition algorithm for these graphs. As an interesting by-product, we find new submatrix characterizations for the well-studied bipartite permutation graphs.

Approximating the path-distance-width for AT-free graphs and graphs in related classes

We consider the problem of determining the path-distance-width for AT-free graphs and graphs in related classes such as k-cocomparability graphs, proper interval graphs, cobipartite graphs, and cochain graphs. We first show that the problem is NP-hard even for cobipartite graphs, and thus for AT-free graphs. Next we present simple approximation algorithms with constant approximation ratios for AT-free graphs and graphs in the related graph classes mentioned above. For instance, our algorithm for AT-free graphs has approximation factor 3 and runs in linear time. We also show that the problem is solvable in polynomial time for cochain graphs, which form a subclass of the class of proper interval graphs.

Polar permutation graphs are polynomial-time recognisable

Polar graphs generalise bipartite graphs, cobipartite graphs, and split graphs, and they constitute a special type of matrix partitions. A graph is polar if its vertex set can be partitioned into two, such that one part induces a complete multipartite graph and the other part induces a disjoint union of complete graphs. Deciding whether a given arbitrary graph is polar, is an NP-complete problem. Here, we show that for permutation graphs this problem can be solved in polynomial time. The result is surprising, as related problems like achromatic number and cochromatic number are NP-complete on permutation graphs. We give a polynomial-time algorithm for recognising graphs that are both permutation and polar. Prior to our result, polarity has been resolved only for chordal graphs and cographs.

On Opsut’s conjecture for hypercompetition numbers of hypergraphs

The notion of the competition hypergraph was introduced as a variant of the notion of the competition graph by Sonntag and Teichert in 2004. They also introduced the notion of the hypercompetition number of a graph.In 1982, Opsut conjectured that for a locally cobipartite graph G, the competition number of G is less than or equal to 2 and the equality holds if and only if the vertex clique cover number of the neighborhood of v is exactly 2 for each vertex v of G. Despite the various attempts to settle the conjecture, it is still open. A hypergraph version of the Opsut’s conjecture can be stated as the assertion that for a hypergraph H, if the number of hyperedges containing v is at most 2 for each vertex v of H, then the hypercompetition number of H is less than or equal to 2 and the equality holds if and only if the number of hyperedges containing v is exactly 2 for each vertex v of H. In this paper, we show that this hypergraph version is true.

Restricted vertex multicut on permutation graphs

Given an undirected graph and pairs of terminals the Restricted Vertex Multicut problem asks for a minimum set of nonterminal vertices whose removal disconnects each pair of terminals. The problem is known to be NP-complete for trees and polynomial-time solvable for interval graphs. In this paper we give a polynomial-time algorithm for the problem on permutation graphs. Furthermore we show that the problem remains NP-complete on split graphs whereas it becomes polynomial-time solvable for the class of co-bipartite graphs.

The competition number of the complement of a cycle

In this paper, we compute the competition number of the complement of a cycle. It is well-known that the competition number of a cycle of length at least 4 is two while the competition number of a cycle of length 3 is one. Characterizing a graph by its competition number has been one of important research problems in the study of competition graphs, and competition numbers of various interesting families of graphs have been found. We thought that it is worthy of computing the competition number of the complement of a cycle. In the meantime, the observation that the complement of an odd cycle of length at least 5 is isomorphic to a circulant graph led us to compute the competition number of a large family of circulant graphs. In fact, those circulant graphs satisfy the long lasting Opsut’s conjecture stating that the competition number of a locally cobipartite graph is at most two.

A Note on the Speed of Hereditary Graph Properties

For a graph property $X$, let $X_n$ be the number of graphs with vertex set $\{1,\ldots,n\}$ having property $X$, also known as the speed of $X$. A property $X$ is called factorial if $X$ is hereditary (i.e. closed under taking induced subgraphs) and $n^{c_1n}\le X_n\le n^{c_2n}$ for some positive constants $c_1$ and $c_2$. Hereditary properties with the speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored, although this family includes many properties of theoretical or practical importance, such as planar graphs or graphs of bounded vertex degree. To simplify the study of factorial properties, we propose the following conjecture: the speed of a hereditary property $X$ is factorial if and only if the fastest of the following three properties is factorial: bipartite graphs in $X$, co-bipartite graphs in $X$ and split graphs in $X$. In this note, we verify the conjecture for hereditary properties defined by forbidden induced subgraphs with at most 4 vertices.

Recognizing line-polar bipartite graphs in time [formula omitted

A graph is polar if the vertex set can be partitioned into A and B in such a way that A induces a complete multipartite graph and B induces a disjoint union of cliques (i.e., the complement of a complete multipartite graph). Polar graphs naturally generalize several classes of graphs such as bipartite graphs, cobipartite graphs and split graphs. Recognizing polar graphs is an NP-complete problem in general, and thus it is of interest to restrict the problem to special classes of graphs. Cographs and chordal graphs are among those whose polarity can be recognized in polynomial time. The line-graphs of bipartite graphs are another class of graphs whose polarity has been characterized recently in terms of forbidden subgraphs, but no polynomial time algorithm is given. In this paper, we present an O ( n ) algorithm which decides whether the line-graph of an input bipartite graph is polar and constructs a polar partition when one exists.

Decompositions for edge-coloring join graphs and cobipartite graphs

An edge-coloring is an association of colors to the edges of a graph, in such a way that no pair of adjacent edges receive the same color. A graph G is Class 1 if it is edge-colorable with a number of colors equal to its maximum degree Δ ( G ) . To determine whether a graph G is Class 1 is NP-complete [I. Holyer, The NP-completeness of edge-coloring, SIAM J. Comput. 10 (1981) 718–720]. First, we propose edge-decompositions of a graph G with the goal of edge-coloring G with Δ ( G ) colors. Second, we apply these decompositions for identifying new subsets of Class 1 join graphs and cobipartite graphs. Third, the proposed technique is applied for proving that the chromatic index of a graph is equal to the chromatic index of its semi-core, the subgraph induced by the maximum degree vertices and their neighbors. Finally, we apply these decomposition tools to a classical result [A.J.W. Hilton, Z. Cheng, The chromatic index of a graph whose core has maximum degree 2, Discrete Math. 101 (1992) 135–147] that relates the chromatic index of a graph to its core, the subgraph induced by the maximum degree vertices.

Sufficient conditions for a graph to be edge-colorable with maximum degree colors

Abstract We study properties of graphs related to the existence of certain vertex and edge partitions. These properties give sufficient conditions for a graph to be Class 1 (i.e., edge-colorable with maximum degree colors). We apply these conditions for solving the classification problem for graphs with acyclic core (the subgraph induced by the maximum degree vertices), and for subclasses of join graphs and cobipartite graphs.

Tree decompositions with small cost

The f-cost of a tree decomposition ( { X i | i ∈ I } , T = ( I , F ) ) for a function f : N → R + is defined as ∑ i ∈ I f ( | X i | ) . This measure associates with the running time or memory use of some algorithms that use the tree decomposition. In this paper, we investigate the problem to find tree decompositions of minimum f-cost. A function f : N → R + is fast, if for every i ∈ N : f ( i + 1 ) ⩾ 2 f ( i ) . We show that for fast functions f, every graph G has a tree decomposition of minimum f-cost that corresponds to a minimal triangulation of G; if f is not fast, this does not hold. We give polynomial time algorithms for the problem, assuming f is a fast function, for graphs that have a polynomial number of minimal separators, for graphs of treewidth at most two, and for cographs, and show that the problem is NP-hard for bipartite graphs and for cobipartite graphs. We also discuss results for a weighted variant of the problem derived of an application from probabilistic networks.