Special Classes Of Graphs Research Articles

On the weak Roman domination number of lexicographic product graphs

A vertex v of a graph G=(V,E) is said to be undefended with respect to a function f:V⟶{0,1,2} if f(v)=0 and f(u)=0 for every vertex u adjacent to v. We call the function f a weak Roman dominating function if for every v such that f(v)=0 there exists a vertex u adjacent to v such that f(u)∈{1,2} and the function f′:V⟶{0,1,2} defined by f′(v)=1, f′(u)=f(u)−1 and f′(z)=f(z) for every z∈V∖{u,v}, has no undefended vertices. The weight of f is w(f)=∑v∈V(G)f(v). The weak Roman domination number of a graph G, denoted by γr(G), is the minimum weight among all weak Roman dominating functions on G. Henning and Hedetniemi (2003) showed that the problem of computing γr(G) is NP-Hard, even when restricted to bipartite or chordal graphs. This suggests finding γr(G) for special classes of graphs or obtaining good bounds on this invariant. In this article, we obtain closed formulae and tight bounds for the weak Roman domination number of lexicographic product graphs in terms of invariants of the factor graphs involved in the product.

The k-diameter component edge connectivity parameter

We focus on a network reliability measure based on edge failures and considering a network operational if there exists a component with diameter k or larger. The k-diameter component edge connectivity parameter of a graph is the minimum number of edge failures needed so that no component has diameter k or larger. This implies each resulting vertex must not have a k-neighbor. We give results for specific graph classes including path graphs, complete graphs, complete bipartite graphs, and a surprising result for perfect r-ary trees.

Deadlock resolution in wait-for graphs by vertex/arc deletion

A deadlock occurs in a distributed computation if a group of processes wait indefinitely for resources from each other. In this paper we study actions to be taken after deadlock detection, especially the action of searching for a small deadlock-resolution set. More precisely, given a “snapshot” graph G representing a deadlocked state of a distributed computation governed by a certain deadlock model $${\mathbb {M}}$$ , we investigate the complexity of vertex/arc deletion problems that aim at finding minimum vertex/arc subsets whose removal turns G into a deadlock-free graph (according to model $${\mathbb {M}}$$ ). Our contributions include polynomial-time algorithms and hardness proofs, for general graphs and for special graph classes. Among other results, we show that the arc deletion problem in the OR model can be solved in polynomial time, and the vertex deletion problem in the OR model remains NP-complete even for graphs with maximum degree $${\varDelta }(G) = 4$$ , but is solvable in $$O (m \sqrt{n})$$ time for graphs with $${\varDelta }(G)\le 3$$ .

Constraint programming models and population-based simulated annealing algorithm for finding graceful and $$\alpha $$ α -labeling of quadratic graphs

In this research, a new mathematical integer programming model is presented for the graph labeling problem of quadratic graphs. The advantages of this model are linearity and the existence of an objective function. Furthermore, two constraint programming models and a meta-heuristics algorithm are also developed to generate feasible graceful labeling and $$\alpha $$ -labeling for special classes of quadratic graphs. Experimental results on large sizes of graphs from the literature show the efficiency of the proposed model and approach.

Rainbow matchings in edge-colored complete split graphs

In 1973, Erdős et al. introduced the anti-Ramsey number for a graph G in Kn, which is defined to be the maximum number of colors in an edge-coloring of Kn which does not contain any rainbow G. This is always regarded as one of rainbow generalizations of the classic Ramsey theory. Since then the anti-Ramsey numbers for several special graph classes in complete graphs have been determined. Also, the researchers generalized the host graph for the anti-Ramsey number from the complete graph to general graphs, including bipartite graphs, complete split graphs, planar graphs, and so on. In this paper, we study the anti-Ramsey number of matchings in the complete split graph. Since the complete split graph contains the complete graph as a subclass, the results in this paper cover the previous results about the anti-Ramsey number of matchings in the complete graph.

Bilevel Model for Adaptive Network Flow Problem

We propose to solve the adaptive network flow problem via a bilevel optimization framework. In this problem, we aim to find a flow that is most robust against any k edges attack. There is an exact algorithm proposed to solve the problem in a specific class of input graphs. However, for some input graphs that are not in that class, a flow obtained from the algorithms is sometimes much less robust than the optimal one. That motivates us to find an efficient exact algorithm based on bilevel optimization framework for the problem in this paper. The framework can give us much better results using reasonable amount of times.

Polynomial and pseudo-polynomial time algorithms for different classes of the Distance Critical Node Problem

We study the Distance Critical Node Problem, a generalisation of the Critical Node Problem where the distances between node pairs impact on the objective function. We establish complexity results for the problem according to specific distance functions and provide polynomial and pseudo-polynomial algorithms for special graph classes such as paths, trees and series–parallel graphs. We also provide additional insights about special cases of the Critical Node Problem variants already tackled in the literature.

Scaffolding Problems Revisited: Complexity, Approximation and Fixed Parameter Tractable Algorithms, and Some Special Cases

This paper is devoted to new results about the scaffolding problem, an integral problem of genome inference in bioinformatics. The problem consists in finding a collection of disjoint cycles and paths covering a particular graph called the “scaffold graph”. We examine the difficulty and the approximability of the scaffolding problem in special classes of graphs, either close to trees, or very dense. We propose negative and positive results, exploring the frontier between difficulty and tractability of computing and/or approximating a solution to the problem. Also, we explore a new direction through related problems consisting in finding a family of edges having a strong effect on solution weight.

Strong matching preclusion number of graphs

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. The strong matching preclusion number (or simply, SMP number) smp(G) of a graph G is the minimum number of vertices and/or edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. This is an extension of the matching preclusion problem and has been introduced by Park and Ihm. In this paper, we first study the SMP number of some special graph classes, and give some sharp upper and lower bounds of SMP number. Next, graphs with large and small SMP number are characterized, respectively. In the end, we investigate the Nordhaus–Gaddum-type relations on SMP number.

On the instability of matching queues

A matching queue is described via a graph, an arrival process and a matching policy. Specifically, to each node in the graph there is a corresponding arrival process of items, which can either be queued or matched with queued items in neighboring nodes. The matching policy specifies how items are matched whenever more than one matching is possible. Given the matching graph and the matching policy, the stability region of the system is the set of intensities of the arrival processes rendering the underlying Markov process positive recurrent. In a recent paper, a condition on the arrival intensities, which was named NCOND, was shown to be necessary for the stability of a matching queue. That condition can be thought of as an analogue to the usual traffic condition for traditional queueing networks, and it is thus natural to study whether it is also sufficient. In this paper, we show that this is not the case in general. Specifically, we prove that, except for a particular class of graphs, there always exists a matching policy rendering the stability region strictly smaller than the set of arrival intensities satisfying NCOND. Our proof combines graph- and queueing-theoretic techniques: After showing explicitly, via fluid-limit arguments that the stability regions of two basic models is strictly included in NCOND, we generalize this result to any graph inducing either one of those two basic graphs.

Some problems on induced subgraphs

We discuss some problems related to induced subgraphs. (1)A good upper bound for the chromatic number in terms of the clique number for graphs in which every induced cycle has length 3 or 4.(2)The perfect chromatic number of a graph, which is the smallest number of perfect sets into which the vertex set of a graph can be partitioned. (A set of vertices is said to be perfect if it induces a perfect graph.)(3)Graphs in which the difference between the chromatic number and the clique number is at most one for every induced subgraph of the graph.(4)A weakening of the notorious Erdős–Hajnal conjecture.(5)A conjecture of Gyárfás about the χ-boundedness of a particular class of graphs.

Bipartite Graphs Associated with 3 Uniform Semigraphs of Trees and its Topological Indices

In this paper, we have studied special class of bipartite graphs associated with 3 uniform semigraphs of path graph \( P_{m,1} \) and star graph \( S_{m,1} \) and estimated some topological indices such as Wiener index, Detour index, Circular index, Cut Circular index, vertex PI index and vertex Co-PI index of these graphs.

Weighted upper domination number

The cardinality of a maximum minimal dominating set of a graph is called its upper domination number. The problem of computing this number is generally NP-hard but can be solved in polynomial time in some restricted graph classes. In this work, we consider the complexity and approximability of the weighted version of the problem in two special graph classes: planar bipartite, split. We also provide an inapproximability result for an unweighted version of this problem in regular graphs.

Note on uniformly transient graphs

We study a special class of graphs with a strong transience feature called uniform transience. We characterize uniform transience via a Feller-type property and via validity of an isoperimetric inequality. We then give a further characterization via equality of the Royden boundary and the harmonic boundary and show that the Dirichlet problem has a unique solution for such graphs. The Markov semigroups and resolvents (with Dirichlet boundary conditions) on these graphs are shown to be ultracontractive. Moreover, if the underlying measure is finite, the semigroups and resolvents are trace class and their generators have \ell^p independent pure point spectra (for 1 \leq p \leq \infty ). Examples of uniformly transient graphs include Cayley graphs of hyperbolic groups as well as trees and Euclidean lattices of dimension at least three. As a surprising consequence, the Royden compactification of such lattices turns out to be the one-point compactification and the Laplacians of such lattices have pure point spectrum if the underlying measure is chosen to be finite.

Solving the minimum edge-dilation k-center problem by genetic algorithms

Two genetic algorithms (GA) for solving the minimum edge-dilation k-center (MEDKC) problem are proposed.Two different representation schemes are based on considering the distance to the central vertices.Specially designed genetic operators mutation and crossover keep individuals feasible.The proposed approaches are comprehensively tested on several classes of graphs.Obtained results demonstrate the efficacy of our proposed algorithms. In this paper, we present two genetic algorithms (GA) for solving the minimum edge-dilation k-center (MEDKC) problem. Up to now, MEDKC has been considered only theoretically, and we present the first heuristic approaches for solving it. In both approaches, the assignment of non-center vertices is based on the distance to the center vertices, while modified crossover and mutation genetic operators, specific for each GA, keep individuals feasible during the entire optimization process. The proposed algorithms were empirically tested and compared by using various sets of differently sized test data: some theoretical classes of graphs with known optimal solutions and some special classes of graphs chosen from the literature for the k-minimum spanning tree and Roman domination problems. Obtained results indicate that newly proposed GAs can be reliably used in constructing routing schemes in complex computer networks.

PBIB-designs and association schemes arising from minimum bi-connected dominating sets of some special classes of graphs

A dominating set D of a connected graph $$G = (V, E)$$ is said to be bi-connected dominating set if the induced subgraphs of both $$\langle D \rangle $$ and $$\langle V-D \rangle $$ are connected. The bi-connected domination number $$\gamma _{bc}(G)$$ is the minimum cardinality of a bi-connected dominating set. A $$\gamma _{bc}$$ -set is a minimum bi-connected dominating set of G. In this paper, we obtain the Partially Balanced Incomplete Block (PBIB)-designs with m = 1, 2, 3, 4 and $$\lfloor \frac{p}{2}\rfloor $$ association schemes arising from $$\gamma _{bc}$$ -sets of some special classes of graphs.

Edge fixed monophonic number of a graph

For an edge xy in a connected graph G of order p ≥ 3, a set SCV(G)is an xy-monophonic set of G if each vertex v Є V(G) lies on an x-u monophonic path or a y-u monophonic path for some element u in S. The minimum cardinality of an xy- monophonic set of G is defined as the xy-monophonic number of G, denoted by mxy (G). An xy-monophonic set of cardinality mxy (G) is called a mxy -set of G. We determine bounds for it and find the same for special classes of graphs. It is shown that for any three positive integers r, d and n ≥ 2 with 2 ≤ r ≤ d, there exists a connected graph G with monophonic radius r, monophonic diameter d and mxy (G) = n for some edge xy in G.

The Upper Connected Vertex Detour Monophonic Number of a Graph

For any vertex x in a connected graph G of order n ≥ 2, a set S x ⊆ V (G) is an x-detour monophonic set of G if each vertex v ∈ V (G) lies on an x-y detour monophonic path for some element y in S x . The minimum cardinality of an x-detour monophonic set of G is the x-detour monophonic number of G, denoted by dm x (G). A connected x-detour monophonic set of G is an x-detour monophonic set S x such that the subgraph induced by S x is connected. The minimum cardinality of a connected x-detour monophonic set of G is the connected x-detour monophonic number of G, denoted by cdm x (G). A connected x-detour monophonic set S x of G is called a minimal connected x-detour monophonic set if no proper subset of S x is a connected x-detour monophonic set. The upper connected x-detour monophonic number of G, denoted by cdm+ x (G), is defined to be the maximum cardinality of a minimal connected x-detour monophonic set of G. We determine bounds and exact values of these parameters for some special classes of graphs. We also prove that for positive integers r,d and k with 2 ≤ r ≤ d and k ≥ 2, there exists a connected graph G with monophonic radius r, monophonic diameter d and upper connected x-detour monophonic number k for some vertex x in G. Also, it is shown that for positive integers j,k,l and n with 2 ≤ j ≤ k ≤ l ≤ n - 3, there exists a connected graph G of order n with dm x (G) = j,dm+ x (G) = k and cdm+ x (G) = l for some vertex x in G.

On the complexity of various parameterizations of common induced subgraph isomorphism

In the Maximum Common Induced Subgraph problem (henceforth MCIS), given two graphs G1 and G2, one looks for a graph with the maximum number of vertices being both an induced subgraph of G1 and G2. MCIS is among the most studied classical NP-hard problems. It remains NP-hard on many graph classes including forests. In this paper, we study the parameterized complexity of MCIS. As a generalization of Clique, it is W[1]-hard parameterized by the size of the solution. Being NP-hard even on forests, most structural parameterizations are intractable. One has to go as far as parameterizing by the size of the minimum vertex cover to get some tractability. Indeed, when parameterized by k:=vc(G1)+vc(G2) the sum of the vertex cover number of the two input graphs, the problem was shown to be fixed-parameter tractable, with an algorithm running in time 2O(klog⁡k). We complement this result by showing that, unless the ETH fails, it cannot be solved in time 2o(klog⁡k). This kind of tight lower bound has been shown for a few problems and parameters but, to the best of our knowledge, not for the vertex cover number. We also show that MCIS does not have a polynomial kernel when parameterized by k, unless NP⊆coNP/poly. Finally, we study MCIS and its connected variant MCCIS on some special graph classes and with respect to other structural parameters.

On ascending subgraph decomposition of graphs

Let G be a graph of positive size q, and let n be that positive integer for which . It was conjectured by Alavi et al that the G can be decomposed into subgraphs G1, G2, … ,Gn such that Gi is isomorphic to a proper subgraph of Gi+1 for 1 ≤ i ≤ n−1. Since the conjecture was posed around 25 years many special classes of graphs have been verified to have ascending subgraph decomposition defined above. Because of the interesting nature of the structure of graphs, many researchers have devoted themselves to work on this problem and also provided several extended versions of such decompositions. Mainly, the size of G and the structure of Gi’s are concerned. As hard as we work on this problem, we find the fascinating inside of this seemingly easy conjecture. This is the reason, we decide to make a survey of this decomposition and hopefully, we would (be very happy to) see the ASD conjecture proved to be true in the near future by some talented researchers.