Special Classes Of Graphs Research Articles

Elegant Fuzzy Labeling of Graphs

Fuzzy graphs and fuzzy labelings are crucial in modeling real-time systems by accommodating varying degrees of information precision. In this paper, we introduce a novel fuzzy labeling called Elegant Fuzzy Labeling. We prove that a simple graph admits Elegant Fuzzy Labeling if and only if it admits elegant labeling. Furthermore, we prove that while a simple graph that admits Elegant Fuzzy Labeling also admits fuzzy labeling, but not conversely. We identify specific classes of simple graphs that admit Elegant Fuzzy Labeling and explore practical application of this new labeling approach.

Basilica: New canonical decomposition in matching theory

AbstractIn matching theory, one of the most fundamental and classical branches of combinatorics, canonical decompositions of graphs are powerful and versatile tools that form the basis of this theory. However, the abilities of the known canonical decompositions, that is, the Dulmage–Mendelsohn, Kotzig–Lovász, and Gallai–Edmonds decompositions, are limited because they are only applicable to particular classes of graphs, such as bipartite graphs, or they are too sparse to provide sufficient information. To overcome these limitations, we introduce a new canonical decomposition that is applicable to all graphs and provides much finer information. This decomposition also provides the answer to the longstanding absence of a canonical decomposition that is nontrivially applicable to general graphs with perfect matchings. We focus on the notion of factor‐components as the fundamental building blocks of a graph; through the factor‐components, our new canonical decomposition states how a graph is organized and how it contains all the maximum matchings. The main results that constitute our new theory are the following: (i) a canonical partial order over the set of factor‐components, which describes how a graph is constructed from its factor‐components; (ii) a generalization of the Kotzig–Lovász decomposition, which shows the inner structure of each factor‐component in the context of the entire graph; and (iii) a canonically described interrelationship between (i) and (ii), which integrates these two results into a unified theory of a canonical decomposition. These results are obtained in a self‐contained way, and our proof of the generalized Kotzig–Lovász decomposition contains a shortened and self‐contained proof of the classical counterpart.

Outer Independent Double Roman Domination Stability in Graphs

An outer independent double Roman dominating function (OIDRDF) on a graph G is a function f : V ( G ) → { 0 , 1 , 2 , 3 } having the property that (i) if f ( v ) = 0 , then the vertex v must have at least two neighbors assigned 2 under f or one neighbor w with f ( w ) = 3 , and if f ( v ) = 1 , then the vertex v must have at least one neighbor w with f ( w ) ≥ 2 and (ii) the subgraph induced by the vertices assigned 0 under f is edgeless. The weight of an OIDRDF is the sum of its function values over all vertices, and the outer independent double Roman domination number γ o i d R ( G ) is the minimum weight of an OIDRDF on G . The γ o i d R -stability ( γ − o i d R -stability, γ + o i d R -stability) of G , denoted by s t γ o i d R ( G ) ( s t − γ o i d R ( G ) , s t + γ o i d R ( G ) ), is defined as the minimum size of a set of vertices whose removal changes (decreases, increases) the outer independent double Roman domination number. In this paper, we determine the exact values on the γ o i d R -stability of some special classes of graphs, and present some bounds on s t γ o i d R ( G ) . In addition, for a tree T with maximum degree Δ , we show that s t γ o i d R ( T ) = 1 and s t − γ o i d R ( T ) ≤ Δ , and characterize the trees that achieve the upper bound.

Dichotomies for Maximum Matching Cut: H-freeness, bounded diameter, bounded radius

The (Perfect) Matching Cut problem is to decide if a graph G has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of G. Both Matching Cut and Perfect Matching Cut are known to be NP-complete. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we perform a complexity study for the Maximum Matching Cut problem, which is to determine a largest matching cut in a graph. Our results yield full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and H-free graphs. A disconnected perfect matching of a graph G is a perfect matching that contains a matching cut of G. We also show how our new techniques can be used for finding a disconnected perfect matching with a largest matching cut for special graph classes. In this way we can prove that the decision problem Disconnected Perfect Matching is polynomial-time solvable for (P6+sP2)-free graphs for every s≥0, extending a known result for P5-free graphs (Bouquet and Picouleau, 2020).

Radio labeling of biconvex split graphs

ABSTRACT A radio labeling of a graph G is a function f : V ( G ) → { 0 , 1 , … } such that for every pair of distinct vertices u , v ∈ V ( G ) , | f ( u ) − f ( v ) | ≥ 1 + diam ( G ) − d ( u , v ) , where diam(G) denotes the diameter of the graph and d(u, v) is the distance between the vertices u and v. The span of a radio labeling f of a graph G is the difference between the least and the largest labels assigned by f and is denoted by, span(f). The radio number of a graph G denoted by, rn(G), is the least positive integer s, such that there exists a radio labeling of G with span s. In this paper, we study the radio labeling of a special class of split graphs called biconvex split graphs of diameter three and we obtain both a lower bound and an upper bound for the radio number of biconvex split graphs of diameter three. Further, we determine the radio number of biconvex split graphs with three maximum degree vertices having disjoint independent neighbors.

Restrained Global Defensive Alliances in Graphs

A defensive alliance in a graph G is a nonempty set of vertices S ⊆ V (G) such that for every vertex v ∈ S, |N[v] ∩ S| ≥ |N(v) ∩ (V (G) ∖ S)|. A defensive alliance S is called global if every vertex in V (G) ∖ S is adjacent to at least one member of the alliance S. In this paper, the concept of restrained global defensive alliance in graphs was introduced. In particular, a global defensive alliance S is a restrained global defensive alliance if the induced subgraph of V ∖ S has no isolated vertex. Here, some properties of this alliance were identified, and its bounds were also determined. In addition, the restrained global defensive alliance number was also formulated, along with characterizations of some special classes of graphs, specifically complete, complete bipartite, and path graphs.

Theoretical studies of the [formula omitted]-strong Roman domination problem

The concept of Roman domination has been a subject of intrigue for more than two decades, with the fundamental Roman domination problem standing out as one of the most significant challenges in this field. This article studies a practically motivated generalization of this problem, known as the k-strong Roman domination. In this problem variant, defenders within a network are tasked with safeguarding any k vertices simultaneously, under multiple attacks. The aim is to find a feasible mapping that assigns a (integer) weight to each vertex of the input graph with a minimum sum of weights across all vertices. A function is considered feasible if any non-defended vertex, i.e. one labeled by zero, is protected by at least one of its neighboring vertices labeled by at least two. Furthermore, each defender ensures the safety of a non-defended vertex by imparting a value of one to it while retaining a one for themselves. To the best of our knowledge, this paper represents the first theoretical study on this problem. The study presents results for general graphs, establishes connections between the problem at hand and other domination problems, and provides exact values and bounds for specific graph classes, including complete graphs, paths, cycles, complete bipartite graphs, grids, and a few selected classes of convex polytopes. Additionally, an attainable lower bound for general cubic graphs is provided.

Leanness Computation: Small Values and Special Graph Classes

Let u and v be vertices in a connected graph G = (V, E). For any integer k such that 0 ≤ k ≤ dG (u, v), the k-slice Sk (u, v) contains all vertices x on a shortest uv-path such that dG (u, x) = k. The leanness of G is the maximum diameter of a slice. This metric graph invariant has been studied under different names, such as "interval thinness" and "fellow traveler property". Graphs with leanness equal to 0, a.k.a. geodetic graphs, also have received special attention in Graph Theory. The practical computation of leanness in real-life complex networks has been studied recently (Mohammed et al., COMPLEX NETWORKS'21). In this paper, we give a finer-grained complexity analysis of two related problems, namely: deciding whether the leanness of a graph G is at most some small value ℓ; and computing the leanness on specific graph classes. We obtain improved algorithms in some cases, and time complexity lower bounds under plausible hypotheses.

A Liouville theorem for elliptic equations with a potential on infinite graphs

We investigate the validity of the Liouville property for a class of elliptic equations with a potential, posed on infinite graphs. Under suitable assumptions on the graph and on the potential, we prove that the unique bounded solution is u≡0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u\\equiv 0$$\\end{document}. We also show that on a special class of graphs the condition on the potential is optimal, in the sense that if it fails, then there exist infinitely many bounded solutions.

Exploring Clique Transversal Problems for d-degenerate Graphs with Fixed d: From Polynomial-Time Solvability to Parameterized Complexity

This paper explores the computational challenges of clique transversal problems in d-degenerate graphs, which are commonly encountered across theoretical computer science and various network applications. We examine d-degenerate graphs to highlight their utility in representing sparse structures and assess several variations of clique transversal problems, including the b-fold and {b}-clique transversal problems, focusing on their computational complexities for different graph categories. Our analysis identifies that certain instances of these problems are polynomial-time solvable in specific graph classes, such as 1-degenerate or 2-degenerate graphs. However, for d-degenerate graphs where d≥2, these problems generally escalate to NP-completeness. We also explore the parameterized complexity, pinpointing specific conditions that render these problems fixed-parameter tractable.

Restrained Global Defensive Alliances on Some Special Classes of Graphs

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On the unicyclic graphs having vertices that belong to all their (strong) metric bases

A metric basis in a graph G is a smallest possible set S of vertices of G, with the property that any two vertices of G are uniquely recognized by using a vector of distances to the vertices in S. A strong metric basis is a variant of metric basis that represents a smallest possible set S′ of vertices of G such that any two vertices x,y of G are uniquely recognized by a vertex v∈S′ by using either a shortest x−v path that contains y, or a shortest y−v path that contains x. Given a graph G, there exist sometimes some vertices of G such that they forcedly belong to every metric basis or to every strong metric basis of G. Such vertices are called (resp. strong) basis forced vertices in G. It is natural to consider finding them, in order to find a (strong) metric basis in a graph. However, deciding about the existence of these vertices in arbitrary graphs is in general an NP-hard problem, which makes desirable the problem of searching for (strong) basis forced vertices in special graph classes. This article centres the attention in the class of unicyclic graphs. It is known that a unicyclic graph can have at most two basis forced vertices. In this sense, several results aimed to classify the unicyclic graphs according to the number of basis forced vertices they have are given in this work. On the other hand, with respect to the strong metric bases, it is proved in this work that unicyclic graphs can have as many strong basis forced vertices as we would require. Moreover, some characterizations of the unicyclic graphs concerning the existence or not of such vertices are given in the exposition as well.

Vertex Spans of Multilayered Cycle and Path Graphs

In this paper, we observe a special class of graphs known as multilayered graphs and their subclasses, namely multilayered cycles and multilayered paths. These graphs model layouts of shopping malls, city street grids, and even resemble the topology of certain famous board games. We analyze the values of all vertex spans (strong, direct, and Cartesian span) for these subclasses of graphs. Surprisingly, our results for multilayered cycles reveal that, regardless of the chosen movement rules, the span values depend solely on the length of the individual cycles, rather than the number of layers. This finding carries significant implications for the application of graph spans in maintaining safety distances.

Abelian Sandpiles on Sierpiński Gasket Graphs

The aim of the current work is to investigate structural properties of the sandpile group of a special class of self-similar graphs. More precisely, we consider Abelian sandpiles on Sierpiński gasket graphs and, for the choice of normal boundary conditions, we give a characterization of the identity element and a recursive description of the sandpile group. Finally, we consider Abelian sandpile Markov chains on the aforementioned graphs and we improve the existing bounds on the speed of convergence to stationarity.

Weak Total Coloring Conjecture and Hadwiger’s Conjecture on Total Graphs

The total graph of a graph $G$, denoted by $T(G)$, is defined on the vertex set $V(G)\cup E(G)$ with $c_1,c_2 \in V(G)\cup E(G)$ adjacent whenever $c_1$ and $c_2$ are adjacent to (or incident on) each other in $G$. The total chromatic number $\chi''(G)$ of a graph $G$ is defined to be the chromatic number of its total graph. The well-known Total Coloring Conjecture or TCC states that for every simple finite graph $G$ having maximum degree $\Delta(G)$, $\chi''(G)\leq \Delta(G) + 2$. In this paper, we consider two ways to weaken TCC:
 (1) Weak TCC: This conjecture states that for a simple finite graph $G$, $\chi''(G) = \chi(T(G)) \leq\Delta(G) + 3$. While weak TCC is known to be true for 4-colorable graphs, it has remained open for 5-colorable graphs. In this paper, we settle this long pending case.
 (2) Hadwiger's Conjecture for total graphs: We can restate TCC as a conjecture that proposes the existence of a strong $\chi$-bounding function for the class of total graphs in the following way: If $H$ is the total graph of a simple finite graph, then $\chi(H) \leq\omega(H) + 1$, where $\omega(H)$ is the clique number of $H$. A natural way to relax this question is to replace $\omega(H)$ by the Hadwiger number $\eta(H)$, the number of vertices in the largest clique minor of $H$. This leads to the Hadwiger's Conjecture (HC) for total graphs: if $H$ is a total graph then $\chi(H) \leq \eta(H)$. We prove that this is true if $H$ is the total graph of a graph with sufficiently large connectivity.
 A second motivation for studying Hadwiger's conjecture for total graphs is the following: Consider the class of split graphs whose vertex set is partitioned into an independent set $A$ and a clique $B$, with the following additional constraints: (1) Each vertex in $B$ has exactly 2 neighbours in $A$; (2) No two vertices in $B$ have the same neighbourhood in $A$. It is known that (European Journal of Combinatorics, 76, 159-174,2019) if Hadwiger's conjecture is proved for the squares of this special class of split graphs, then it holds also for the general case. Of course, proving the conjecture for this specialzed-looking case is indeed difficult since it is only a reformulation of the general case, and therefore it is natural to consider the difficulty level of Hadwiger's conjecture for the squares of graph classes defined by slighly modifying the above class of graphs. A natural structural modification is to assume that both $A$ and $B$ are independent sets, keeping everything else same. It turns out that the squares of this modified class of graphs is exactly the class of total graphs. From this perspective, it is not really surprising that HC on Total Graphs is also challenging. On the other hand, we show that weak TCC implies HC on total graphs. This perhaps suggests that the latter is an easier problem than the former.

Contrastive fine-tuning for low-resource graph-level transfer learning

Due to insufficient supervision and the gap between pre-training pretext tasks and downstream tasks, transferring pre-trained graph neural networks (GNNs) to downstream tasks in low-resource scenarios remains challenging. In this paper, a Contrastive Fine-tuning (Con-tuning) framework is proposed for low-resource graph-level transfer learning, and a graph-level supervised contrastive learning (SCL) task is designed within the framework as the first attempt to introduce SCL for fine-tuning processes of pre-trained GNNs. The SCL task compensates for the insufficient supervision problem in low-resource scenarios and narrows the gap between pretext tasks and downstream tasks. To further reinforce the supervision signal in the SCL task, we devise a graphon theory based labeled graph generator to extract the generalized knowledge of a specific class of graphs. Based on this knowledge, graph-level templates are generated for each class and used as contrastive samples in the SCL task. Then, the proposed Con-tuning framework jointly learns the SCL task and downstream tasks to effectively fine-tune the pre-trained GNNs for downstream tasks. Extensive experiments with eight real-world datasets show that Con-tuning framework enables pre-trained GNNs to achieve better performance on graph-level downstream tasks in low-resource settings.

2-Domination Number for Special Classes of Hypercubes, Enhanced Hypercubes and Knodel Graphs

2-Domination Number for Special Classes of Hypercubes, Enhanced Hypercubes and Knodel Graphs

On odd colorings of sparse graphs

An oddc-coloring of a graph is a proper c-coloring such that each non-isolated vertex has a color appearing an odd number of times within its open neighborhood. A proper conflict-freec-coloring of a graph is a proper c-coloring such that each non-isolated vertex has a color appearing exactly once within its neighborhood. Clearly, every proper conflict-free c-coloring is also an odd c-coloring. Cranston conjectured that every graph G with maximum average degree mad(G)<4cc+2 (where c≥4) has an odd c-coloring, and he proved this conjecture for c∈{5,6}. Note that the bound 4cc+2 is best possible. Cho et al. solved Cranston’s conjecture for c≥5, strengthening the result by transitioning from odd c-coloring to proper conflict-free c-coloring. However, they did not provide all the extremal non-colorable graphs G with mad(G)=4cc+2, which remains an open question of interest.In this paper, we tackle this intriguing extremal problem. We aim to characterize all non-proper conflict-free c-colorable graphs G with mad(G)=4cc+2. For the case of c=4, Cranston’s conjecture is not true, as evidenced by the existence of a counterexample: a graph whose every block is a 5-cycle. Cho et al. proved that a graph G with mad(G)<229 and no induced 5-cycles has an odd 4-coloring. We improve this result by proving that a graph G with mad(G)≤229 (with equality allowed) is not odd 4-colorable if and only if G belongs to a specific class of graphs. On the other hand, Cho et al. established that a planar graph with girth at least 5 has an odd 6-coloring; we improve it by proving that a planar graph without 4−-cycles adjacent to 7−-cycles also has an odd 6-coloring.

Anti-Ramsey number of matchings in outerplanar graphs

A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. Given a graph family G and a graph H, the anti-Ramsey number AR(G,H) is the maximum number of colors in an edge-coloring of a graph G∈G which has no rainbow copy of H. When G={G}, we write AR(G,H) for short. Erdős, Simonovits and Sós introduced these numbers in the 1970s, they studied the case when G is a complete graph. Since then, many results have appeared considering G and H belonging to specific graph classes. In particular, Jendrol’, Schiermeyer and Tu (Jendrol’ et al., 2014) obtained an upper bound for AR(Tn,kK2) in terms of n and k for k≥3 and n≥2k, where Tn is the family of maximal planar graphs on n vertices. Jin and Ye (Jin and Ye, 2018) continued the study of anti-Ramsey numbers of matchings in planar graphs, by deriving an upper bound for AR(Mn,kK2), where Mn is the family of maximal outerplanar graphs on n vertices. In both works, examples are shown for which there are colorings using many colors while avoiding rainbow matchings, but the numbers of colors in these examples do not reach the upper bounds. The bounds of AR(Tn,kK2) were improved in following works by other researchers. This paper goes in the same direction of these two works, proving exact value for AR(Bn,kK2) in terms of n and k for n≥6 and k≥3, where Bn is the family of maximal bipartite outerplanar graphs on n vertices which contains a Hamilton cycle.

Minimal toughness in special graph classes

Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest $t$ for which the graph is $t$-tough, whereby the toughness of complete graphs is defined as infinity. A graph is minimally $t$-tough if the toughness of the graph is $t$, and the deletion of any edge from the graph decreases the toughness. In this paper, we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the classes of chordal graphs, split graphs, claw-free graphs, and $2K_2$-free graphs.