Showing papers on "Split graph published in 2021"

Threshold graphs under picture Dombi fuzzy information

Abstract: The aggregation feature, decision-making skills and operational characteristics of multi-purpose Dombi operators make them a highly adaptable tool for compiling the imprecise information. This study exploits the generalized structure of Dombi operators and significant characteristics of picture fuzzy sets $$(\mathcal {PFS}_{s})$$ to extend the theory of fuzzy graph by presenting the premium concept of picture Dombi fuzzy threshold graphs $$(\mathcal {PDFTG}_{s}).$$ We prove that $$\mathcal {PDFTG}_{s}$$ do not induce picture Dombi fuzzy alternating $$(\mathcal {PDFA})$$ 4-cycle as induced subgraph, and these graphs can be constructed periodically by adding an isolated or dominant vertex to a single vertex graph. We demonstrate that $$\mathcal {PDFTG}_{s}$$ are triangulated graphs. We show that the crisp graph of $$\mathcal {PDFTG}$$ is a split graph $$({\mathcal {S}}{\mathcal {G}})$$ . Further, we illustrate the notion of threshold dimension and threshold partition number of picture Dombi fuzzy graphs $$(\mathcal {PDFG}_{s})$$ . Moreover, we present some fundamental results related to threshold dimension and threshold partition number with the appropriate illustration. Finally, we discuss the implementation of $$\mathcal {PDFTG}_{s}$$ in the distribution of coal resources.

Word-representability of split graphs generated by morphisms

Abstract: A graph $G=(V,E)$ is word-representable if and only if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$, $x eq y$, alternate in $w$ if and only if $xy\in E$. A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. There is a long line of research on word-representable graphs in the literature, and recently, word-representability of split graphs has attracted interest. In this paper, we first give a characterization of word-representable split graphs in terms of permutations of columns of the adjacency matrices. Then, we focus on the study of word-representability of split graphs obtained by iterations of a morphism, the notion coming from combinatorics on words. We prove a number of general theorems and provide a complete classification in the case of morphisms defined by $2\times 2$ matrices.

On the geodetic hull number for complementary prisms II

Abstract: In the geodetic convexity, a set of vertices S of a graph G is convex if all vertices belonging to any shortest path between two vertices of S lie in S . The convex hull H (S ) of S is the smallest convex set containing S . If H (S ) = V (G ), then S is a hull set . The cardinality h (G ) of a minimum hull set of G is the hull number of G . The complementary prism GG of a graph G arises from the disjoint union of the graph G and G by adding the edges of a perfect matching between the corresponding vertices of G and G . A graph G is autoconnected if both G and G are connected. Motivated by previous work, we study the hull number for complementary prisms of autoconnected graphs. When G is a split graph, we present lower and upper bounds showing that the hull number is unlimited. In the other case, when G is a non-split graph, it is limited by 3.

Semi-transitivity of directed split graphs generated by morphisms

Abstract: A directed graph is semi-transitive if and only if it is acyclic and for any directed path $u_1\rightarrow u_2\rightarrow \cdots \rightarrow u_t$, $t \geq 2$, either there is no edge from $u_1$ to $u_t$ or all edges $u_i\rightarrow u_j$ exist for $1 \leq i < j \leq t$. In this paper, we study semi-transitivity of families of directed split graphs obtained by iterations of morphisms applied to the adjacency matrices and giving in the limit infinite directed split graphs. A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. We fully classify semi-transitive infinite directed split graphs when a morphism in question can involve any $n\times m$ matrices over $\{-1,0,1\}$ with a single natural condition.

The Locating Chromatic Number for Split Graph of Cycle

Abstract: The minimum number of colors in a locating coloring of G is called the locating chromatic number of graph G, denoted by

Forbidden subgraphs in reduced power graphs of finite groups

Abstract: Let $ G $ be a finite group. The reduced power graph of $ G $ is the undirected graph whose vertex set consists of all elements of $ G $, and two distinct vertices $ x $ and $ y $ are adjacent if either $ \langle x\rangle \subset \langle y\rangle $ or $ \langle y\rangle \subset \langle x\rangle $. In this paper, we show that the reduced power graph of $ G $ is perfect and characterize all finite groups whose reduced power graphs are split graphs, cographs, chordal graphs, and threshold graphs. We also give complete classifications in the case of abelian groups, dihedral groups, and generalized quaternion groups.

On (distance) signless Laplacian spectra of graphs

Abstract: Let Q(G), $${{\mathcal {D}}(G)}$$ and $${{\mathcal {D}}}^Q(G)={{\mathcal {D}}iag(Tr)} + {{\mathcal {D}}(G)}$$ be, respectively, the signless Laplacian matrix, the distance matrix and the distance signless Laplacian matrix of graph G, where $${{\mathcal {D}}iag(Tr)}$$ denotes the diagonal matrix of the vertex transmissions in G. The eigenvalues of Q(G) and $${{\mathcal {D}}}^Q(G)$$ will be denoted by $$q_{1} \ge q_{2} \ge \cdots \ge q_{n-1} \ge q_n$$ and $$\partial ^Q_1 \ge \partial ^Q_2 \ge \cdots \ge \partial ^Q_{n-1} \ge \partial ^Q_n$$ , respectively. A graph G which does not share its distance signless Laplacian spectrum with any other non-isomorphic graphs is said to be determined by its distance signless Laplacian spectrum. Characterizing graphs with respect to spectra of graph matrices is challenging. In literature, there are many graphs that are proved to be determined by the spectra of some graph matrices (adjacency matrix, Laplacian matrix, signless Laplacian matrix, distance matrix etc.). But there are much fewer graphs that are proved to be determined by the distance signless Laplacian spectrum. Namely, the path graph, the cycle graph, the complement of the path and the complement of the cycle are proved to be determined by the distance signless Laplacian spectra. In this paper, we establish Nordhaus–Gaddum-type results for the least signless Laplacian eigenvalue of graph G. Moreover, we prove that the join graph $$G\vee K_{q}$$ is determined by the distance singless Laplacian spectrum when G is a $$p-2$$ regular graph of order p. Finally, we show that the short kite graph and the complete split graph are determined by the distance signless Laplacian spectra. Our approach for characterizing these graphs with respect to distance signless Laplacian spectra is different from those given in literature.

Efficiency and stability in the connections model with heterogeneous nodes

Abstract: This paper studies the connections model (Jackson and Wolinsky, 1996) when nodes may have different values. It is shown that efficiency is reached by a strongly hierarchical structure that we call strong NSG-networks: Nested Split Graph networks where the hierarchy or ranking of nodes inherent in any such network is consistent with the rank of nodes according to their value, perhaps leaving some of the nodes with the lowest values disconnected. A simple algorithm is provided for calculating these efficient networks. We also introduce a natural extension of pairwise stability assuming that players are allowed to agree on how the cost of each link is split and prove that stability in this sense for connected strong NSG-networks entails efficiency.

A Simple \((2+\epsilon )\)-Approximation Algorithm for Split Vertex Deletion

Abstract: A split graph is a graph whose vertex set can be partitioned into a clique and a stable set. Given a graph G and weight function \(w: V(G) \rightarrow \mathbb {Q}_{\ge 0}\), the Split Vertex Deletion (SVD) problem asks to find a minimum weight set of vertices X such that \(G-X\) is a split graph. It is easy to show that a graph is a split graph if and only if it does not contain a 4-cycle, 5-cycle, or a two edge matching as an induced subgraph. Therefore, SVD admits an easy 5-approximation algorithm. On the other hand, for every \(\delta >0\), SVD does not admit a \((2-\delta )\)-approximation algorithm, unless P = NP or the Unique Games Conjecture fails.

An infinite class of Neumaier graphs and non-existence results

Abstract: A Neumaier graph is a non-complete edge-regular graph containing a regular clique. A Neumaier graph that is not strongly regular is called a strictly Neumaier graph. In this work we present a new construction of strictly Neumaier graphs, and using Jacobi sums, we show that our construction produces infinitely many instances. Moreover, we prove some necessary conditions for the existence of (strictly) Neumaier graphs that allow us to show that several parameter sets are not admissible.

A Note on Coloring $(4K_1, C_4, C_6)$-free graphs with a $C_7$

Abstract: Even-hole-free graphs are a graph class of much interest. Foley et al. [Graphs Comb. 36(1): 125-138 (2020)] have recently studied $(4K_1, C_4, C_6)$-free graphs, which form a subclass of even-hole-free graphs. Specifically, Foley et al. have shown an algorithm for coloring these graphs via bounded clique-width if they contain a $C_7$. In this note, we give a simpler and much faster algorithm via a more restrictive graph parameter, neighborhood diversity.

On clique numbers of colored mixed graphs.

Abstract: An (m,n)-colored mixed graph, or simply, an (m,n)-graph is a graph having m different types of arcs and n different types of edges. A homomorphism of an (m,n)-graph G to another (m,n)-graph H is a vertex mapping that preserves adjacency, the type thereto and the direction. A subset R of the set of vertices of G that always maps distinct vertices in itself to distinct image vertices under any homomorphism is called an (m,n)-relative clique of G. The maximum cardinality of an (m,n)-relative clique of a graph is called the (m,n)-relative clique number of the graph. In this article, we explore the (m,n)-relative clique numbers for various families of graphs.

Toughness, 2-factors and Hamiltonian cycles in $2K_2$-free graphs

Abstract: A graph $G$ is called a $2K_2$-free graph if it does not contain $2K_2$ as an induced subgraph. In 2014, Broersma et al. showed that every 25-tough $2K_2$-free graph with at least three vertices is Hamiltonian. Recently, Shan improved this result by showing that 3-tough is sufficient instead of 25-tough. On the other hand, Kratsch et al. showed that for any $t <\frac{3}{2}$ there exists a $t$-tough split graph without 2-factors (also, it is a $2K_2$-free graph). In this paper, we present two results. First, we show that every $\frac{3}{2}$-tough $2K_2$-free graph has a $2$-factor. This result is sharp by the result in split graphs. Second, we show that every 2-tough $2K_2$-free graph is Hamiltonian, which was conjectured by Mou and Pasechnik.

Eternal Domination and Clique Covering

Abstract: We study the relationship between the eternal domination number of a graph and its clique covering number. Using computational methods, we show that the smallest graph having its eternal domination number less than its clique covering number has $10$ vertices. This answers a question of Klostermeyer and Mynhardt [Protecting a graph with mobile guards, Appl. Anal. Discrete Math. $10$ $(2016)$, no. $1$, $1-29$]. We also determine the complete set of $10$-vertex and $11$-vertex graphs having eternal domination numbers less than their clique covering numbers. In addition, we study the problem on triangle-free graphs, circulant graphs, planar graphs and cubic graphs. Our computations show that all triangle-free graphs and all circulant graphs of order $12$ or less have eternal domination numbers equal to their clique covering numbers, and exhibit $13$ triangle-free graphs and $2$ circulant graphs of order $13$ which do not have this property. Using these graphs, we describe a method to generate an infinite family of triangle-free graphs and an infinite family of circulant graphs with eternal domination numbers less than their clique covering numbers. Our computations also show that all planar graphs of order $11$ or less, all $3$-connected planar graphs of order $13$ or less and all cubic graphs of order less than $18$ have eternal domination numbers equal to their clique covering numbers. Finally, we show that for any integer $k \geq 2$ there exist infinitely many graphs having domination number and eternal domination number equal to $k$ containing dominating sets which are not eternal dominating sets. This answers another question of Klostermeyer and Mynhardt [Eternal and Secure Domination in Graphs, Topics in domination in graphs, Dev. Math. $64$ $(2020)$, $445-478$, Springer, Cham].

On semi-transitive orientability of split graphs

Abstract: A directed graph is semi-transitive if and only if it is acyclic and for any directed path $u_1\rightarrow u_2\rightarrow \cdots \rightarrow u_t$, $t \geq 2$, either there is no edge from $u_1$ to $u_t$ or all edges $u_i\rightarrow u_j$ exist for $1 \leq i < j \leq t$. Recognizing semi-transitive orientability of a graph is an NP-complete problem. A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Semi-transitive orientability of spit graphs was recently studied in the literature. The main result in this paper is proving that recognition of semi-transitive orientability of split graphs can be done in a polynomial time. We also characterize, in terms of minimal forbidden induced subgraphs, semi-transitively orientable split graphs with the size of the independent set at most 3, hence extending the known classification of such graphs with the size of the clique at most 5.

Characterizations for split graphs and unbalanced split graphs.

Abstract: We introduce a characterization for split graphs by using edge contraction. Then, we use it to prove that any ($2K_{2}$, claw)-free graph with $\alpha(G) \geq 3$ is a split graph. Also, we apply it to characterize any pseudo-split graph. Finally, by using edge contraction again, we characterize unbalanced split graphs which we use to characterize the Nordhaus-Gaddum graphs.

Chip-Firing on the Complete Split Graph: Motzkin Words and Tiered Parking Functions

Abstract: We highlight some results from studying chip-firing on the the complete split graph [5]. In this work it is shown that recurrent states can be characterised in terms of Motzkin words and can also be characterised in terms of a new type of parking function that we call a tiered parking function. These new parking functions arise by assigning a tier (or colour) to each of the cars, and specifying how many cars of a lower-tier one wishes to have parked before them.

On the clique behavior of graphs of low degree

Abstract: To any simple graph $G$, the clique graph operator $K$ associates the graph $K(G)$ which is the intersection graph of the maximal complete subgraphs of $G$. The iterated clique graphs are defined by $K^{0}(G)=G$ and $K^{n}(G)=K(K^{n-1}(G))$ for $n\geq 1$. If there are $m