Class Of Bipartite Graphs Research Articles

On a class of column‐weight 3 decomposable LDPC codes with the analysis of elementary trapping sets

AbstractA column‐weight LDPC code with the parity‐check matrix is called decomposable if there exists a permutation on the rows of , such that can be decomposed into column‐weight one matrix. In this paper, some variations of edge coloring of graphs are used to construct some column‐weight three decomposable LDPC codes with girths at least six and eight. Applying the presented method on several known classes of bipartite graphs, some classes of column‐weight three decomposable LDPC codes are derived having flexibility in length and rate. Interestingly, the constructed parity‐check matrices based on the proper edge coloring of graphs can be considered as the base matrix of some high rate column‐weight three quasi‐cyclic (QC) LDPC codes with maximum‐achievable girth 20. The paper also leads to a simple characterization of elementary trapping sets of the decomposable codes based on the chromatic index of the corresponding normal graphs. This characterization corresponds to a simple search algorithm finds all possible existing elementary trapping sets in a girth‐6 or girth‐8 column‐weight 3 LDPC code which are layered super set of a short cycle in the Tanner graph of the code. Simulation results indicate that the QC‐LDPC codes with large girths lifted from the constructed base matrices have good performances over AWGN channel.

Bipartite (P6,C6)-Free Graphs: Recognition and Optimization Problems

The canonical decomposition of a bipartite graph is a new decomposition method that involves three operators: parallel, series, and K⨁ S. The class of weak-bisplit graphs is the class of totally decomposable graphs with respect to these operators, and the class of bicographs is the class of totally decomposable graphs with respect to parallel and series operators. We prove in this paper that the class of bipartite (P6,C6)-free graphs is the class of bipartite graphs that are totally decomposable with respect to parallel and K⨁S operators. We present a linear time recognition algorithm for (P6,C6)-free graphs that is symmetrical to the linear recognition algorithms of weak-bisplit graphs and star1,2,3-free bipartite graphs. As a result of this algorithm, we present efficient solutions in this class of graphs for two optimization graph problems: the maximum balanced biclique problem and the maximum independent set problem.

Excluding a planar matching minor in bipartite graphs

The notion of matching minors is a specialisation of minors fit for the study of graphs with perfect matchings. Matching minors have been used to give a structural description of bipartite graphs on which the number of perfect matchings can be computed efficiently, based on a result of Little, by McCuaig et al. in 1999.In this paper we generalise basic ideas from the graph minor series by Robertson and Seymour to the setting of bipartite graphs with perfect matchings. We introduce a version of Erdős-Pósa property for matching minors and find a direct link between this property and planarity. From this, it follows that a class of bipartite graphs with perfect matchings has bounded perfect matching width if and only if it excludes a planar matching minor. We also present algorithms for bipartite graphs of bounded perfect matching width for a matching version of the disjoint paths problem, matching minor containment, and for counting the number of perfect matchings. From our structural results, we obtain that recognising whether a bipartite graph G contains a fixed planar graph H as a matching minor, and that counting the number of perfect matchings of a bipartite graph that excludes a fixed planar graph as a matching minor are both polynomial time solvable.

Role coloring bipartite graphs

A k -role coloring α of a graph G is an assignment of k colors to the vertices of G such that every color is used at least once and if any two vertices are assigned the same color, then their neighborhood are assigned the same set of colors. By definition, every graph on n vertices admits an n -role coloring. While for every graph on n vertices, it is trivial to decide if it admits a 1-role coloring, determining whether a graph admits a k -role coloring is a notoriously hard problem for k ≥ 2 . In fact, it is known that k - Role coloring is NP -complete for k ≥ 2 on general graph class. There has been extensive research on the complexity of k - role coloring on various hereditary graph classes. Furthering this direction of research, we show that k - Role coloring is NP -complete on bipartite graphs for k ≥ 3 (while it is trivial for k = 2 ). We complement the hardness result by characterizing 3-role colorable bipartite chain graphs, leading to a polynomial time algorithm for 3- Role coloring for this class of graphs. We further show that 2- Role coloring is NP -complete for graphs that are d vertices or edges away from the class of bipartite graphs, even when d = 1 .

Classes of nonbipartite graphs with reciprocal eigenvalue property

Let G be a simple connected graph and A(G) be the adjacency matrix of G. The graph G is said to have the reciprocal eigenvalue property (R) if A(G) is nonsingular and 1λ is an eigenvalue of A(G) whenever λ is an eigenvalue of A(G). Further, if λ and 1λ have the same multiplicity, for each eigenvalue λ, then G is said to have the strong reciprocal eigenvalue property (SR). Till date, all the classes of bipartite graphs that are found to have property (R), are found to have property (SR) and it is not known whether these two properties are equivalent even for the bipartite graphs with a unique perfect matching. Among nonbipartite graphs, there is only one known graph class for which these two properties are not equivalent. In this article, we construct some more classes of nonbipartite graphs with property (R) but not (SR).

On minimizing the maximum color for the 1–2–3 Conjecture

The 1–2–3 Conjecture asserts that, for every connected graph different from K2, its edges can be labeled with 1,2,3 so that, when coloring each vertex with the sum of its incident labels, no two adjacent vertices get the same color. This conjecture takes place in the more general context of distinguishing labelings, where the goal is to label graphs so that some pairs of their elements are distinguishable relatively to some parameter computed from the labeling.In this work, we investigate the consequences of labeling graphs as in the 1–2–3 Conjecture when it is further required to make the maximum resulting color as small as possible. In some sense, we aim at producing a number of colors that is as close as possible to the chromatic number of the graph. We first investigate the hardness of determining the minimum maximum color by a labeling for a given graph, which we show is NP-complete in the class of bipartite graphs but polynomial-time solvable in the class of graphs with bounded treewidth. We then provide bounds on the minimum maximum color that can be generated both in the general context, and for particular classes of graphs. Finally, we study how using larger labels permit to reduce the maximum color.

On caterpillar factors in graphs

A caterpillar is either a K2 or a tree on at least 3 vertices such that deleting its leaves we obtain a path of order at least 1. Given a simple undirected graph G=(V,E), a caterpillar factor of G is a set of caterpillar subgraphs of G such that each vertex v∈V belongs to exactly one of them. A caterpillar factor F is internally even if every vertex of degree degF(v)≥2 has an even degree; F is odd if degF(v) is odd for every v∈V(G). We present a linear-time algorithm that decides whether a tree admits an internally even caterpillar factor and, on the other hand, we prove that the decision problem is NP-complete on the class of planar bipartite graphs. For the odd caterpillar factor problem, we obtain similar results. It can be decided in linear time over the class of trees, but the problem is NP-complete on the class of bipartite graphs.

The firebreak problem

AbstractSuppose we have a network that is represented by a graph G. Potentially a fire (or other type of contagion) might erupt at some vertex of G. We are able to respond to this outbreak by establishing a firebreak at k other vertices of G, so that the fire cannot pass through these fortified vertices. The question that now arises is which k vertices will result in the greatest number of vertices being saved from the fire, assuming that the fire will spread to every vertex that is not fully behind the k vertices of the firebreak. This is the essence of the Firebreak decision problem, which is the focus of this paper. We establish that the problem is intractable on the class of split graphs as well as on the class of bipartite graphs, but can be solved in linear time when restricted to graphs having constant‐bounded treewidth, or in polynomial time when restricted to intersection graphs. We also consider some closely related problems.

Perfect Italian domination on planar and regular graphs

A perfect Italian dominating function of a graph G=(V,E) is a function f:V→{0,1,2} such that for every vertex f(v)=0, it holds that ∑u∈N(v)f(u)=2, i.e., the weight of the labels assigned by f to the neighbors of v is exactly 2. The weight of a perfect Italian function is the sum of the weights of the vertices. The perfect Italian domination number of G, denoted by γIp(G), is the minimum weight of any perfect Italian dominating function of G. While introducing the parameter, Haynes and Henning (2019) also proposed the problem of determining the best possible constants cG such that γIp(G)≤cG×n for all graphs of order n when G is in a particular class G of graphs. They proved that cG=1 when G is the class of bipartite graphs, and raised the question for planar graphs and regular graphs. We settle their question precisely for planar graphs by proving that cG=1 and for cubic graphs by proving that cG=2∕3. For split graphs, we also show that cG=1. In addition, we characterize the graphs G with γIp(G) equal to 2 and 3 and determine the exact value of the parameter for several simple structured graphs. We conclude by proving that it is NP-complete to decide whether a given bipartite planar graph admits a perfect Italian dominating function of weight k.

Cooperative Colorings of Trees and of Bipartite Graphs


 Given a system $(G_1, \ldots ,G_m)$ of graphs on the same vertex set $V$, a cooperative coloring is a choice of vertex sets $I_1, \ldots ,I_m$, such that $I_j$ is independent in $G_j$ and $\bigcup_{j=1}^{m}I_j = V$. For a class $\mathcal{G}$ of graphs, let $m_{\mathcal{G}}(d)$ be the minimal $m$ such that every $m$ graphs from $\mathcal{G}$ with maximum degree $d$ have a cooperative coloring. We prove that $\Omega(\log\log d) \le m_\mathcal{T}(d) \le O(\log d)$ and $\Omega(\log d)\le m_\mathcal{B}(d) \le O(d/\log d)$, where $\mathcal{T}$ is the class of trees and $\mathcal{B}$ is the class of bipartite graphs.

Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1‐Sperner hypergraphs

AbstractA hypergraph is said to be 1‐Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of ‐Sperner hypergraphs to graphs. First, we consider several ways of associating hypergraphs to graphs, namely, vertex cover, clique, independent set, dominating set, and closed neighborhood hypergraphs. For each of them, we characterize graphs yielding ‐Sperner hypergraphs. These results give new characterizations of threshold and domishold graphs. Second, we apply a characterization of ‐Sperner hypergraphs to derive decomposition theorems for two classes of split graphs, a class of bipartite graphs, and a class of cobipartite graphs. These decomposition theorems, based on certain matrix partitions, lead to new classes of graphs of bounded clique‐width and new polynomially solvable cases of three basic domination problems: domination, total domination, and connected domination.

Quasimonotone graphs

For any class C of bipartite graphs, we define quasi-C to be the class of all graphs G such that every bipartition of G belongs to C. This definition is motivated by a generalisation of the switch Markov chain on perfect matchings from bipartite graphs to nonbipartite graphs. The monotone graphs, also known as bipartite permutation graphs and proper interval bigraphs, are such a class of bipartite graphs. We investigate the structure of quasi-monotone graphs and hence construct a polynomial time recognition algorithm for graphs in this class.

Permanental generating functions and sequential importance sampling

We introduce techniques for deriving closed form generating functions for enumerating permutations with restricted positions keeping track of various statistics. The method involves evaluating permanents with variables as entries. These are applied to determine the sample size required for a novel sequential importance sampling algorithm for generating random perfect matchings in classes of bipartite graphs.

English

We study lower estimates for integral fuctionals for which the structure of the integrand is defined by a graph, in particular, by a bipartite graph. Functionals of such kind appear in statistical mechanics and quantum chemistry in the context of Mayer’s transformation and Mayer’s cluster integrals. Integral functionals generated by graphs play an important role in the theory of graph limits. Specific kind of functionals generated by bipartite graphs are at the center of the famous and much studied Sidorenko’s conjecture, where a certain lower bound is conjectured to hold for every bipartite graph. In the present paper we work with functionals more general and lower bounds significantly sharper than those in Sidorenko’s conjecture. In his 1991 seminal paper, Sidorenko proved such sharper bounds for several classes of bipartite graphs. To obtain his result he used a certain way of “gluing” graphs. We prove his inequality for a new class of bipartite graphs by defining a different type of gluing.

Bipartite and Eulerian minors

This paper defines the related notions of bipartite and Eulerian minors for binary matroids. Using these definitions, it characterizes graphic matroids within the classes of bipartite binary matroids and Eulerian binary matroids by the exclusion of certain bipartite minors and Eulerian minors, respectively. This result on Eulerian minors in binary matroids extends a result of Chudnovsky et al. who characterized planar graphs within the class of bipartite graphs by the exclusion of K3,3 as a bipartite minor.

On the barrier graph of an arrangement of ray sensors

Wireless sensor networks are commonly used to monitor various environmental conditions. Possible geometries for the region covered by a sensor include disks, wedges, and rays, among others, depending on the function of the sensor. In this paper we consider a network consisting of ray sensors deployed to detect intruders traversing a path, not necessarily straight, from a source α to a destination β. The coverage of the network with respect to α and β can be modeled by a tripartite graph, the barrier graph of the network. While all barrier graphs are tripartite, the converse is not true (for instance, C5 is not a barrier graph).The main result of this paper is a rigidity theorem on the structure of barrier graphs that results from constraints imposed by the geometry of the network. This allows us to show that almost all bipartite graphs are not barrier graphs, despite the fact that various classes of bipartite graphs, including trees, cycles of even length, and Km,n are barrier graphs. Furthermore, vertex cover of a barrier graph corresponds to a set of sensors whose removal allows a clear path from α to β. While all bipartite graphs with small vertex covers are barrier graphs (a fact we prove for sizes less than 4), the rigidity property also implies that graphs with vertex covers bigger than a certain constant are not barrier graphs.

On the inverse of a class of weighted graphs

In this article, only connected bipartite graphs $G$ with a unique perfect matching $\c{M}$ are considered. Let $G_\w$ denote the weighted graph obtained from $G$ by giving weights to its edges using the positive weight function $\w:E(G)\ar (0,\ity)$ such that $\w(e)=1$ for each $e\in\c{M}$. An unweighted graph $G$ may be viewed as a weighted graph with the weight function $\w\equiv\1$ (all ones vector). A weighted graph $G_\w$ is nonsingular if its adjacency matrix $A(G_\w)$ is nonsingular. The {\em inverse} of a nonsingular weighted graph $G_\w$ is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix $A(G_\w)$ via a diagonal matrix whose diagonal entries are either $1$ or $-1$. In [S.K.~Panda and S.~Pati. On some graphs which possess inverses. {\em Linear and Multilinear Algebra}, 64:1445--1459, 2016.], the authors characterized a class of bipartite graphs $G$ with a unique perfect matching such that $G$ is invertible. That class is denoted by $\c{H}_{nmc}$. It is natural to ask whether $G_\w$ is invertible for each invertible graph $G\in\c{H}_{nmc}$ and for each weight function $\w\not\equiv\1$. In this article, first an example is given to show that there is an invertible graph $G\in\c{H}_{nmc}$ and a weight function$\w\not\equiv\1$ such that $G_\w$ is not invertible. Then the weight functions $\w$ for each graph $G\in\c{H}_{nmc}$ such that $G_\w$ is invertible, are characterized.

Solving the Maximum Matching Problem on Bipartite <i>Star</i><sub>123</sub>-Free Graphs in Linear Time

The bipartite Star123-free graphs were introduced by V. Lozin in [1] to generalize some already known classes of bipartite graphs. In this paper, we extend to bipartite Star123-free graphs a linear time algorithm of J. L. Fouquet, V. Giakoumakis and J. M. Vanherpe for finding a maximum matching in bipartite Star123, P7-free graphs presented in [2]. Our algorithm is a solution of Lozin’s conjecture.

Matching-Star Ramsey Minimal Graphs

Let G and H be any graphs without isolated vertices. The Ramsey set \({\mathcal{R}(G,H)}\) consists of all graphs F without isolated vertices such that \({F \rightarrow (G,H)}\) and \({F-e \nrightarrow (G,H)}\) for every \({e \in E(F)}\). In this paper, we give necessary and sufficient conditions of graphs belonging to the set \({\mathcal{R}(3K_2,K_{1,n})}\), for any n ≥ 3. Furthermore, by using computational approach, we determine all Ramsey \({(3K_2,K_{1,n})}\)-minimal graphs of order at most 10 vertices for \({3 \leq n \leq 7}\). We construct two classes of bipartite graphs in \({\mathcal{R}(3K_2,K_{1,n})}\), for any \({n \geq 3}\). We also present a class of graphs containing a clique of six vertices in this set. We give large classes of graphs in the set \({\mathcal{R}(t K_2, K_{1,n})}\), for \({n \geq 3}\) and \({t \geq 4}\), that are constructed from t disjoint stars.

The bipartite unicyclic graphs with the first [formula omitted] largest matching energies

The theory of matching energy of graphs since been proposed by Gutman and Wagner in 2012, has attracted more and more attention. It is denoted by Bn,m, the class of bipartite graphs with order n and size m. In particular, Bn,n denotes the set of bipartite unicyclic graphs, which is an interesting class of graphs. In this paper, for odd n, we characterize the bipartite unicyclic graphs with the first ⌊n−34⌋ largest matching energies. There is an interesting correspondence: we conclude that the graph with the second maximal matching energy in Bn,n for odd n ≥ 11 is Pn6, which is the only graph attaining the maximum value of the energy among all the (bipartite) unicyclic graphs for n ≥ 16.