Containment relations in split graphs

A graph containment problem is to decide whether one graph can be modified into some other graph by using a number of specified graph operations. We consider edge deletions, edge contractions, vertex deletions and vertex dissolutions as possible graph operations permitted. By allowing any combination of these four operations we capture the following ten problems: testing on (induced) minors, (induced) topological minors, (induced) subgraphs, (induced) spanning subgraphs, dissolutions and contractions. A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. Our results combined with existing results settle the parameterized complexity of all ten problems for split graphs.     

On the NP-hardness of edge-deletion and -contraction problems

If π is a property on graphs, the corresponding edge deletion (edge contraction, respectively) problem is: Given a graph G, determine the minimum number of edges of G whose deletion (contraction) results in a graph satisfying property π. We show that these problems are NP-hard if π is finitely characterizable by 3-connected graphs.     

On the removal of forbidden graphs by edge-deletion or by edge-contraction

If π is a property on graphs, the corresponding edge deletion (edge contraction, respectively) problem is: Given a graph G, determine the minimum number of edges of G whose deletion (contraction) results in a graph satisfying property π. We show that these problems are NP-hard if π is finitely characterizable by 3-connected graphs.     

Cohesion and stability in graphs

The cohesion of a vertex v is the minimum number of edges whose deletion makes v a cutvertex of the resulting graph. Such considerations are particularly interesting for alliance and friendship graphs. In this paper, we further examine cohesion, first introduced in [3], with our major goal here being the examination of stability under edge addition. We find necessary and sufficient conditions for a vertex to be stable and prove a theorem which discovers total vertex stability in several graphs.     

Oriented bipartite graphs and the Goldbach graph

In this paper, we study oriented bipartite graphs. In particular, we introduce “bitransitive” graphs. Several characterizations of bitransitive bitournaments are obtained. We show that bitransitive bitounaments are equivalent to acyclic bitournaments. As applications, we characterize acyclic bitournaments with Hamiltonian paths, determine the number of non-isomorphic acyclic bitournaments of a given order, and solve the graph-isomorphism problem in linear time for acyclic bitournaments. Next, we prove the well-known Caccetta-Häggkvist Conjecture for oriented bipartite graphs in some cases for which it is unsolved, in general, for oriented graphs. We also introduce the concept of undirected as well as oriented “odd-even” graphs. We characterize bipartite graphs and acyclic oriented bipartite graphs in terms of them. In fact, we show that any bipartite graph (acyclic oriented bipartite graph) can be represented by some odd-even graph (oriented odd-even graph). We obtain some conditions for connectedness of odd-even graphs. This study of odd-even graphs and their connectedness is motivated by a special family of odd-even graphs which we call “Goldbach graphs”. We show that the famous Goldbach's conjecture is equivalent to the connectedness of Goldbach graphs. Several other number theoretic conjectures (e.g., the twin prime conjecture) are related to various parameters of Goldbach graphs, motivating us to study the nature of vertex-degrees and independent sets of these graphs. Finally, we observe Hamiltonian properties of some odd-even graphs related to Goldbach graphs for a small number of vertices.     

Computing the bipartite edge frustration of fullerene graphs

Bipartite edge frustration of a graph is defined as the smallest number of edges that have to be deleted from the graph to obtain a bipartite spanning subgraph. We show that for fullerene graphs this quantity can be computed in polynomial time and obtain explicit formulas for the icosahedral fullerenes. We also report some computational results and discuss a potential application of this invariant in the context of fullerene stability.     

Edge Partition of Graphs Embeddable in the Projective Plane and the Klein Bottle

In a previous paper by the author joint with Baogang XU published in Discrete Math in 2018, we show that every non-planar toroidal graph can be edge partitioned into a planar graph and an outerplanar graph. This edge partition then implies some results in thickness and outerthickness of toroidal graphs. In particular, if each planar graph has outerthickness at most $2$ (conjectured by Chartrand, Geller and Hedetniemi in 1971 and the confirmation of the conjecture was announced by Gon\c{c}alves in 2005), then the outerthickness of toroidal graphs is at most 3 which is the best possible due to $K_7$. In this paper we continue to study the edge partition for projective planar graphs and Klein bottle embeddable graphs. We show that (1) every non-planar but projective planar graph can be edge partitioned into a planar graph and a union of caterpillar trees; and (2) every non-planar Klein bottle embeddable graph can be edge partitioned into a planar graph and a subgraph of two vertex amalgamation of a caterpillar tree with a cycle with pendant edges. As consequences, the thinkness of projective planar graphs and Klein bottle embeddabe graphs are at most $2$, which are the best possible, and the outerthickness of these graphs are at most $3$.     

Classification of interpolation theorems for spanning trees and other families of spanning subgraphs

We say that a graphical invariant i of a graph interpolates over a family F of graphs if i satisfies the following property: If m and M are the minimum and maximum values (respectively) of i over all graphs in F then for each k, m ? k ? M, there is a graph H in F for which i(H)= k. In previous works it was shown that when F is the set of spanning trees of a connected graph G, a large number of invariants interpolate (some of these invariants require the additional assumption that G be 2-connected). Although the proofs of all these results use the same basic idea of gradually transforming one tree into another via a sequence of edge exchanges, some of these processes require sequences that use more properties of trees than do others. We show that the edge exchange proofs can be divided into three types, in accordance with the extent to which the exchange sequence depends upon properties of spanning trees. This idea is then used to obtain new interpolation results for some invariants, and to show how the exchange methods and interpolation results on spanning trees can be extended to other families of spanning subgraphs.     

On Generating Triangle-Free Graphs

We show that the problem to decide whether a graph can be made triangle-free with at most k edge deletions remains NP-complete even when restricted to planar graphs of maximum degree seven. In addition, we provide polynomial-time data reduction rules for this problem and obtain problem kernels consisting of 6k vertices for general graphs and 11k/3 vertices for planar graphs.     

Matching preclusion and conditional matching preclusion for regular interconnection networks

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those induced by a single vertex. It is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices and neither perfect matchings nor almost-perfect matchings. In this paper, we prove general results regarding the matching preclusion number and the conditional matching preclusion number as well as the classification of their respective optimal sets for regular graphs. We then use these general results to study the problems for Cayley graphs generated by 2-trees and the hyper Petersen networks.     

The Proportional Colouring Problem: Optimizing Buffers in Wireless Mesh Networks

In this paper, we consider a new edge colouring problem motivated by wireless mesh networks optimization: the proportional edge colouring problem. Given a graph G with positive weights associated to its edges, we want to find a proper edge colouring which assigns to each edge at least a proportion (given by its weight) of all the colours. If such colouring exists, we want to find one using the minimum number of colours. We proved that deciding if a weighted graph admits a proportional edge colouring is polynomial while determining its proportional edge chromatic number is NP-hard. We also give a lower and an upper bound that can be polynomially computed. We finally characterize some graphs and weighted graphs for which we can determine the proportional edge chromatic number.     

On perturbations of almost distance-regular graphs

In this paper we show that certain almost distance-regular graphs, the so-called h-punctually walk-regular graphs, can be characterized through the cospectrality of their perturbed graphs. A graph G with diameter D is called h-punctually walk-regular, for a given h≤D, if the number of paths of length ? between a pair of vertices u,v at distance h depends only on ?. The graph perturbations considered here are deleting a vertex, adding a loop, adding a pendant edge, adding/removing an edge, amalgamating vertices, and adding a bridging vertex. We show that for walk-regular graphs some of these operations are equivalent, in the sense that one perturbation produces cospectral graphs if and only if the others do. Our study is based on the theory of graph perturbations developed by Cvetkovi?, Godsil, McKay, Rowlinson, Schwenk, and others. As a consequence, some new characterizations of distance-regular graphs are obtained.     

Bidegreed graphs are edge reconstructible

An edge-deleted subgraph of a graph G is a subgraph obtained from G by the deletion of an edge. The Edge Reconstruction Conjecture asserts that every simple finite graph with four or more edges is determined uniquely, up to isomorphism, by its collection of edge-deleted subgraphs. A class of graphs is said to be edge reconstructible if there is no graph in the class with four or more edges that is not edge reconstructible. This paper proves that bidegreed graphs (graphs whose vertices all have one of two possible degrees) are edge reconstructible. The results are then generalized to show that all graphs that do not have three consecutive integers in their degree sequence are also edge reconstructible.     

Constructably Laplacian integral graphs

A graph is Laplacian integral if the spectrum of its Laplacian matrix consists entirely of integers. We consider the class of constructably Laplacian integral graphs - those graphs that be constructed from an empty graph by adding a sequence of edges in such a way that each time a new edge is added, the resulting graph is Laplacian integral. We characterize the constructably Laplacian integral graphs in terms of certain forbidden vertex-induced subgraphs, and consider the number of nonisomorphic Laplacian integral graphs that can be constructed by adding a suitable edge to a constructably Laplacian integral graph. We also discuss the eigenvalues of constructably Laplacian integral graphs, and identify families of isospectral nonisomorphic graphs within the class.     

On 2-switches and isomorphism classes

A 2-switch is an edge addition/deletion operation that changes adjacencies in the graph while preserving the degree of each vertex. A well-known result states that graphs with the same degree sequence may be changed into each other via sequences of 2-switches. We show that if a 2-switch changes the isomorphism class of a graph, then it must take place in one of four configurations. We also present a sufficient condition for a 2-switch to change the isomorphism class of a graph. As consequences, we give a new characterization of matrogenic graphs and determine the largest hereditary graph family whose members are all the unique realizations (up to isomorphism) of their respective degree sequences.     

Incrementally extensible hypercube networks and their fault tolerance

The incrementally extensible hypercube or IEH graph, a generalization of the hypercube, was introduced in [1]. It is shown that IEH graphs are incrementally extensible in steps of 1, optimally fault tolerant and its diameter is logarithmic in the number of nodes. Moreover, for any given number of nodes, the difference of the maximum and the minimum degree of a node in the graph is ≤ 1; i.e., the graph is almost regular. We also analyze these graphs in presence of node failures. We develop a fault tolerant routing algorithm which can compute a minimal path between two arbitrary nodes in an IEH graph in the presence of the maximum number of allowable faults. We also determine the fault diameter of the graph. Thus, we show that the IEH graphs, while they can be defined for any integer number of computing nodes, retain all the desirable properties of regular hypercubes including fault tolerance and a tight fault diameter.     

On Spin Models,Triply Regular Association Schemes,and Duality

Motivated by the construction of invariants of links in 3-space, we study spin models on graphs for which all edge weights (considered as matrices) belong to the Bose-Mesner algebra of some association scheme. We show that for series-parallel graphs the computation of the partition function can be performed by using series-parallel reductions of the graph appropriately coupled with operations in the Bose-Mesner algebra. Then we extend this approach to all plane graphs by introducing star-triangle transformations and restricting our attention to a special class of Bose-Mesner algebras which we call exactly triply regular. We also introduce the following two properties for Bose-Mesner algebras. The planar duality property (defined in the self-dual case) expresses the partition function for any plane graph in terms of the partition function for its dual graph, and the planar reversibility property asserts that the partition function for any plane graph is equal to the partition function for the oppositely oriented graph. Both properties hold for any Bose-Mesner algebra if one considers only series-parallel graphs instead of arbitrary plane graphs. We relate these notions to spin models for link invariants, and among other results we show that the Abelian group Bose-Mesner algebras have the planar duality property and that for self-dual Bose-Mesner algebras, planar duality implies planar reversibility. We also prove that for exactly triply regular Bose-Mesner algebras, to check one of the above properties it is sufficient to check it on the complete graph on four vertices. A number of applications, examples and open problems are discussed.     

NP‐completeness of list coloring and precoloring extension on the edges of planar graphs

In the edge precoloring extension problem, we are given a graph with some of the edges having preassigned colors and it has to be decided whether this coloring can be extended to a proper k‐edge‐coloring of the graph. In list edge coloring every edge has a list of admissible colors, and the question is whether there is a proper edge coloring where every edge receives a color from its list. We show that both problems are NP‐complete on (a) planar 3‐regular bipartite graphs, (b) bipartite outerplanar graphs, and (c) bipartite series‐parallel graphs. This improves previous results of Easton and Parker 6 , and Fiala 8 . © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 313–324, 2005     

Changing and unchanging domination: a classification

The six classes of graphs resulting from the changing or unchanging of the domination number of a graph when a vertex is deleted, or an edge is deleted or added are considered. Each of these classes has been studied individually in the literature. We consider relationships among the classes, which are illustrated in a Venn diagram. We show that no subset of the Venn diagram is empty for arbitrary graphs, and prove that some of the subsets are empty for connected graphs. Our main result is a characterization of trees in each subset of the Venn diagram.     

On critically perfect graphs

A perfect graph is critical, if the deletion of any edge results in an imperfect graph. We give examples of such graphs and prove some basic properties. We relate critically perfect graphs to well-known classes of perfect graphs, investigate the structure of the class of critically perfect graphs, and study operations preserving critical perfectness. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 394–404, 1999