Special Classes Of Graphs Research Articles

Graft Analogue of general Kotzig–Lovász decomposition

Canonical decompositions, such as the Gallai–Edmonds and Dulmage–Mendelsohn decompositions, are a series of structure theorems that form the foundation of matching theory. The classical Kotzig–Lovász decomposition is a canonical decomposition applicable to a special class of graphs with perfect matchings, called factor-connected graphs, and is famous for its contribution to the studies of perfect matching polytopes and lattices. Recently, this decomposition has been generalized for general graphs with perfect matchings; this generalized decomposition is called the general Kotzig–Lovász decomposition . In fact, this generalized decomposition can be presented as a component of a more composite canonical decomposition called the Basilica decomposition . As such, the general Kotzig–Lovász decomposition has contributed to the derivation of various new results in matching theory, such as an alternative proof of the tight cut lemma and a characterization of maximal barriers in general graphs. Joins in grafts, also known as T -joins in graphs, is a classical concept in combinatorics and is a generalization of perfect matchings in terms of parity. More precisely, minimum joins and grafts are generalizations of perfect matchings and graphs with perfect matchings, respectively. Under the analogy between matchings and joins, analogues of the canonical decompositions for grafts are expected to be strong and fundamental tools for studying joins. In this paper, we provide a generalization of the general Kotzig–Lovász decomposition for arbitrary grafts. Our result also contains Sebö’s generalization of the classical Kotzig–Lovász decomposition for factor-connected grafts. From our results in this paper, a generalization of the Dulmage–Mendelsohn decomposition, which is originally a classical canonical decomposition for bipartite graphs, has been obtained for bipartite grafts. This paper is the first of a series of papers that establish a generalization of the Basilica decomposition for grafts.

On the edge chromatic vertex stability number of graphs

For an arbitrary invariant of a graph G, the vertex stability number is the minimum number of vertices of G whose removal results in a graph with or with In this paper, first we give some general lower and upper bounds for the ρ-vertex stability number, and then study the edge chromatic vertex stability number of graphs, where is edge chromatic number (chromatic index) of G. We prove some general results for this parameter and determine for specific classes of graphs.

Towards Obtaining a 3-Decomposition From a Perfect Matching

A decomposition of a graph is a set of subgraphs whose edges partition those of $G$. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph, and a matching. It has been settled for special classes of graphs, one of the first results being for Hamiltonian graphs. In the past two years several new results have been obtained, adding the classes of plane, claw-free, and 3-connected tree-width 3 graphs to the list.
 In this paper, we regard a natural extension of Hamiltonian graphs: removing a Hamiltonian cycle from a cubic graph leaves a perfect matching. Conversely, removing a perfect matching $M$ from a cubic graph $G$ leaves a disjoint union of cycles. Contracting these cycles yields a new graph $G_M$. The graph $G$ is star-like if $G_M$ is a star for some perfect matching $M$, making Hamiltonian graphs star-like. We extend the technique used to prove that Hamiltonian graphs satisfy the 3-decomposition conjecture to show that 3-connected star-like graphs satisfy it as well.

On disjoint maximum and maximal independent sets in graphs and inverse independence number

In this paper, we give a class of graphs that do not admit disjoint maximum and maximal independent (MMI) sets. The concept of inverse independence was introduced by Bhat and Bhat in [Inverse independence number of a graph, Int. J. Comput. Appl. 42(5) (2012) 9–13]. Let [Formula: see text] be a [Formula: see text]-set in [Formula: see text]. An independent set [Formula: see text] is called an inverse independent set with respect to [Formula: see text]. The inverse independence number [Formula: see text] is the size of the largest inverse independent set in [Formula: see text]. Bhat and Bhat gave few bounds on the independence number of a graph, we continue the study by giving some new bounds and exact value for particular classes of graphs: spider tree, the rooted product and Cartesian product of two particular graphs.

Token shifting on graphs

ABSTRACT We investigate a new variation of a token reconfiguration problem on graphs using the cyclic shift operation. A coloured or labelled token is placed on each vertex of a given graph, and a ‘move’ consists in choosing a cycle in the graph and shifting tokens by one position along its edges. Given a target arrangement of tokens on the graph, our goal is to find a shortest sequence of moves that will re-arrange the tokens as in the target arrangement. The novelty of our model is that tokens are allowed to shift along any cycle in the graph, as opposed to a given subset of its cycles. We first discuss the problem on special graph classes: we give efficient algorithms for optimally solving the 2-Coloured Token Shifting Problem on complete graphs and block graphs, as well as the Labelled Token Shifting Problem on complete graphs and variants of barbell graphs. We then show that, in the 2-Coloured Token Shifting Problem, the shortest sequence of moves is NP-hard to approximate within a factor of , even for grid graphs. The latter result settles an open problem posed by Sai et al.

Finding Hamiltonian cycles of truncated rectangular grid graphs in linear time

The Hamiltonian cycle problem is an important problem in graph theory. For solid grid graphs, an O(n4)-time algorithm has been given. In this paper, we solve the problem for a special class of solid grid graphs, i.e. truncated rectangular grid graphs, in linear time.

Algorithms for maximum internal spanning tree problem for some graph classes

For a given graph G, a maximum internal spanning tree of G is a spanning tree of G with maximum number of internal vertices. The Maximum Internal Spanning Tree (MIST) problem is to find a maximum internal spanning tree of the given graph. The MIST problem is a generalization of the Hamiltonian path problem. Since the Hamiltonian path problem is NP-hard, even for bipartite and chordal graphs, two important subclasses of graphs, the MIST problem also remains NP-hard for these graph classes. In this paper, we propose linear-time algorithms to compute a maximum internal spanning tree of cographs, block graphs, cactus graphs, chain graphs and bipartite permutation graphs. The optimal path cover problem, which asks to find a path cover of the given graph with maximum number of edges, is also a well studied problem. In this paper, we also study the relationship between the number of internal vertices in maximum internal spanning tree and number of edges in optimal path cover for the special graph classes mentioned above.

Sharp conditions on fractional ID-(g, f)-factor-critical covered graphs

Combining the concept of a fractional (g, f)-covered graph with that of a fractional ID-(g, f)-factor-critical graph, we define the concept of a fractional ID-(g, f)-factor-critical covered graph. This paper reveals the relationship between some graph parameters and the existence of fractional ID-(g, f)-factor-critical covered graphs. A sufficient condition for a graph being a fractional ID-(g, f)-factor-critical covered graph is presented. In addition, we demonstrate the sharpness of the main result in this paper by constructing a special graph class. Furthermore, the relationship between other graph parameters(such as binding number, toughness, sun toughness and neighborhood union) and fractional ID-(g, f)-factor-critical covered graphs can be studied further.

More on the outer connected geodetic number of a graph

For a connected graph [Formula: see text] of order at least two, a total outer connected geodetic set [Formula: see text] of a graph [Formula: see text] is an outer connected geodetic set such that the subgraph induced by [Formula: see text] has no isolated vertices. The minimum cardinality of a total outer connected geodetic set of [Formula: see text] is the total outer connected geodetic number of [Formula: see text] and is denoted by [Formula: see text]. We determine bounds for it and also find the total outer connected geodetic number for some special classes of graphs. It is shown that for positive integers [Formula: see text] and [Formula: see text] with [Formula: see text] there exists a connected graph [Formula: see text] with [Formula: see text] [Formula: see text] and [Formula: see text]. It is proved that for each triple [Formula: see text] and [Formula: see text] of positive integers with [Formula: see text], [Formula: see text] and [Formula: see text], there exists a connected graph [Formula: see text] of order [Formula: see text] such that [Formula: see text] and [Formula: see text]. It is also shown that for positive integers [Formula: see text] such that [Formula: see text] with [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the outer connected geodetic number of a graph [Formula: see text].

Complexity of the Operation Graphs on Cycle and Complete Graph

The number of spanning trees (or complexity) of a graph can be used to measure the reliabilities of network and circuit design. Although many NP hard problems are related to the complexity of graphs, there are explicit formulas for the complexity of some specific graph classes. This paper mainly studies the explicit expression of the complexity of graphs obtained by graph operations. Through classical graph operations, we obtain some operation graphs generated by cycle and complete graph, and get the closed formulas for the complexity in these operation graphs. Compared with Daoud’s results, we obtain some interesting algebraic properties about the complexity of operation graphs.

Many Faces of Symmetric Edge Polytopes

Symmetric edge polytopes are a class of lattice polytopes constructed from finite simple graphs. In the present paper we highlight their connections to the Kuramoto synchronization model in physics — where they are called adjacency polytopes — and to Kantorovich-Rubinstein polytopes from finite metric space theory. Each of these connections motivates the study of symmetric edge polytopes of particular classes of graphs. We focus on such classes and apply algebraic combinatorial methods to investigate invariants of the associated symmetric edge polytopes.

On the signed strong total Roman domination number of graphs

Let $G=(V,E)$ be a finite and simple graph of order $n$ and maximumdegree $\Delta$. A signed strong total Roman dominating function ona graph $G$ is a function $f:V(G)\rightarrow\{-1, 1,2,\ldots, \lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the condition that (i) forevery vertex $v$ of $G$, $f(N(v))=\sum_{u\in N(v)}f(u)\geq 1$, where$N(v)$ is the open neighborhood of $v$ and (ii) every vertex $v$ forwhich $f(v)=-1$ is adjacent to at least one vertex$w$ for which $f(w)\geq 1+\lceil\frac{1}{2}\vert N(w)\cap V_{-1}\vert\rceil$, where$V_{-1}=\{v\in V: f(v)=-1\}$.The minimum of thevalues $\omega(f)=\sum_{v\in V}f(v)$, taken over all signed strongtotal Roman dominating functions $f$ of $G$, is called the signed strong totalRoman domination number of $G$ and is denoted by $\gamma_{ssTR}(G)$.In this paper, we initiate signed strong total Roman domination number of a graph and giveseveral bounds for this parameter. Then, among other results, we determine the signed strong total Roman dominationnumber of special classes of graphs.

Partitioning H-free graphs of bounded diameter

A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of H-free graphs, that is, graphs that do not contain some graph H as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph H contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study (MFCS 2019) on the k-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the H-free input graph. We continue this line of research by initiating a complexity study on the impact of bounding the diameter for a variety of classical vertex partitioning problems restricted to H-free graphs. We prove that bounding the diameter does not help for Independent Set, but leads to new tractable cases for problems closely related to 3-Colouring. That is, we show that Near-Bipartiteness, Independent Feedback Vertex Set, Independent Odd Cycle Transversal, Acyclic 3-Colouring and Star 3-Colouring are all polynomial-time solvable for chair-free graphs of bounded diameter. To obtain these results we exploit a new structural property of 3-colourable chair-free graphs.

Partitioning the edge set of a bipartite graph into the minimal number of subgraphs isomorphic to those of a simple 4 order cycle

In this paper, we study the computational complexity for a problem of partitioning the edge set of a bipartite graph into the minimal number of subgraphs isomorphic to those of a simple cycle of order 4 in special graph classes. This problem is NP-hard and finds application in organizing the distribution of network packets over communication channels in the process of transmission from one router to another. We develop anO(nlogn) algorithm for solving that problem in a class of n order trees. Intractable cases of the problem are identified.

Parameterized Complexity of Geodetic Set

A vertex set $S$ of a graph $G$ is geodetic if every vertex of $G$ lies on a shortest path between two vertices in $S$. Given a graph $G$ and $k \in \mathbb{N}$, the NP-hard ${\rm G{\small EODETIC}~S{ \small ET}}$ problem asks whether there is a geodetic set of size at most $k$. Complementing various works on ${\rm G{\small EODETIC}~S{ \small ET}}$ restricted to special graph classes, we initiate a parameterized complexity study of ${\rm G{\small EODETIC}~S{ \small ET}}$ and show, on the one side, that ${\rm G{\small EODETIC}~S{ \small ET}}$ is $W[1]$-hard when parameterized by feedback vertex number, path-width, and solution size, combined. On the other side, we develop fixed-parameter algorithms with respect to the feedback edge number, the tree-depth, and the modular-width of the input graph.

On the Restrained Cost Eective Sets of Some Special Classes of Graphs

Let G be a nontrivial, undirected, simple graph. Let S be a subset of V (G). S is a restrained cost effective set of G if for each vertex v in S, degS(v) \(\leq\) degV (G)rS(v) and the subgraph induced by the vertex set, V (G) r S has no isolated vertex. The maximum cardinality of a restrained cost effective set is the restrained cost effective number, CEr(G). In this paper, the restrained cost effective sets of paths, cycles, complete graphs, complete product of graphs and graphs resulting from line graph of graphs with maximum degree of 2 were characterized. As a direct consequence, the bounds or exact values for the restrained cost effective number were determined as well.

The adjacency dimension of graphs

It is known that the problem of computing the adjacency dimension of a graph is NP-hard. This suggests finding the adjacency dimension for special classes of graphs or obtaining good bounds on this invariant. In this work we obtain general bounds on the adjacency dimension of a graph G in terms of known parameters of G . We discuss the tightness of these bounds and, for some particular classes of graphs, we obtain closed formulae. In particular, we show the close relationships that exist between the adjacency dimension and other parameters, like the domination number, the location-domination number, the 2 -domination number, the independent 2 -domination number, the vertex cover number, the independence number and the super domination number.

On a List Variant of the Multiplicative 1-2-3 Conjecture

The 1-2-3 Conjecture asks whether almost all graphs can be (edge-)labelled with 1, 2, 3 so that no two adjacent vertices are incident to the same sum of labels. In the last decades, several aspects of this problem have been studied in literature, including more general versions and slight variations. Notable such variations include the List 1-2-3 Conjecture variant, in which edges must be assigned labels from dedicated lists of three labels, and the Multiplicative 1-2-3 Conjecture variant, in which labels 1, 2, 3 must be assigned to the edges so that adjacent vertices are incident to different products of labels. Several results obtained towards these two variants led to observe some behaviours that are distant from those of the original conjecture. In this work, we consider the list version of the Multiplicative 1-2-3 Conjecture, proposing the first study dedicated to this very problem. In particular, given any graph G, we wonder about the minimum k such that G can be labelled as desired when its edges must be assigned labels from dedicated lists of size k. Exploiting a relationship between our problem and the List 1-2-3 Conjecture, we provide upper bounds on k when G belongs to particular classes of graphs. We further improve some of these bounds through dedicated arguments.

The local complement metric dimension of graphs

The metric dimension of graphs has been extended in some types and variations such as local metric dimension and complement metric dimension of graphs. Merging two concepts is one way of developing the concept of graph theory as a branch of mathematics. These two variations motivated us to construct a new concept of metric dimension so called local complement metric dimension. We apply this new concept to some particular classes of graphs and the corona product of two graphs. We also characterize the local complement metric dimension of graphs with certain properties, namely, bipartite graphs and odd cycle graphs. Furthermore, we discover the local complement metric dimension of the corona product of two particular graphs as well as the corona product of two general graphs.

Towards improving Christofides algorithm on fundamental classes by gluing convex combinations of tours

We present a new approach for gluing tours over certain tight, 3-edge cuts. Gluing over 3-edge cuts has been used in algorithms for finding Hamilton cycles in special graph classes and in proving bounds for 2-edge-connected subgraph problem, but not much was known in this direction for gluing connected multigraphs. We apply this approach to the traveling salesman problem (TSP) in the case when the objective function of the subtour elimination relaxation is minimized by a \(\theta \)-cyclic point: \(x_e \in \{0,\theta , 1-\theta , 1\}\), where the support graph is subcubic and each vertex is incident to at least one edge with x-value 1. Such points are sufficient to resolve TSP in general. For these points, we construct a convex combination of tours in which we can reduce the usage of edges with x-value 1 from the \(\frac{3}{2}\) of Christofides algorithm to \(\frac{3}{2}-\frac{\theta }{10}\) while keeping the usage of edges with fractional x-value the same as Christofides algorithm. A direct consequence of this result is for the Uniform Cover Problem for TSP: In the case when the objective function of the subtour elimination relaxation is minimized by a \(\frac{2}{3}\)-uniform point: \(x_e \in \{0, \frac{2}{3}\}\), we give a \(\frac{17}{12}\)-approximation algorithm for TSP. For such points, this lands us halfway between the approximation ratios \(\frac{3}{2}\) of Christofides algorithm and \(\frac{4}{3}\) implied by the famous “four-thirds conjecture”.