Disjoint Union Of Complete Graphs Research Articles

On two variants of split graphs: 2-unipolar graph and k-probe-split graph

A graph is called split if its vertex set can be partitioned into a stable set and a clique. In this article, we studied two variants of split graphs. A graph G is polar if its vertex set can be partitioned into two sets A and B such that G[A] is a complete multipartite graph and G[B] is a disjoint union of complete graphs. A 2-unipolar graph is a polar graph G such that G[A] is a clique and G[B] is the disjoint union of complete graphs with at most two vertices. We present a minimal forbidden induced subgraph characterization for 2-unipolar graphs. In addition, we show that they can be represented as an intersection of substars of special cacti. Let G be a graph class, the G-width of a graph G is the minimum positive integer k such that there exist k independent sets N1, … , Nk such that a set F of nonedges of G, whose endpoints belong to some Ni with i = 1, … , k, can be added so that the resulting graph G′ belongs to G. We say that a graph G is k-probe-G if it has G-width at most k and when G is the class of split graphs it is denominated k-probe-split. We prove that deciding, given a graph G and a positive integer k, whether G is a h-probe-split graph for some h ≤ k is NP-complete. Besides, a characterization by minimal forbidden induced subgraphs for 2-probe-split cographs is presented.

On local antimagic total labeling of complete graphs amalgamation

Let \(G = (V,E)\) be a connected simple graph of order \(p\) and size \(q\). A graph \(G\) is called local antimagic (total) if \(G\) admits a local antimagic (total) labeling. A bijection \(g : E \to \{1,2,\ldots,q\}\) is called a local antimagic labeling of $ if for any two adjacent vertices \(u\) and \(v\), we have \(g^+(u) \ne g^+(v)\), where \(g^+(u) = \sum_{e\in E(u)} g(e)\), and \(E(u)\) is the set of edges incident to \(u\). Similarly, a bijection \(f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}\) is called a local antimagic total labeling of \(G\) if for any two adjacent vertices \(u\) and \(v\), we have \(w_f(u)\ne w_f(v)\), where \(w_f(u) = f(u) + \sum_{e\in E(u)} f(e)\). Thus, any local antimagic (total) labeling induces a proper vertex coloring of \(G\) if vertex \(v\) is assigned the color \(g^+(v)\) (respectively, \(w_f(u)\)). The local antimagic (total) chromatic number, denoted \(\chi_{la}(G)\) (respectively \(\chi_{lat}(G)\)), is the minimum number of induced colors taken over local antimagic (total) labeling of \(G\). In this paper, we determined \(\chi_{lat}(G)\) where \(G\) is the amalgamation ofcomplete graphs. Consequently, we also obtained the local antimagic (total) chromatic number of the disjoint union of complete graphs, and the join of \(K_1\) and amalgamation of complete graphs under various conditions. An application of local antimagic total chromatic number is also given.

Laplacian spectra of power graphs of certain prime-power Abelian groups

The power graph [Formula: see text] of a semigroup [Formula: see text] is a simple undirected graph whose vertex set is [Formula: see text] itself, and any two distinct vertices are adjacent if one of them is a power of the other. In this paper, we describe the power graph [Formula: see text] in terms of joins and disjoint unions of complete graphs, and use this to calculate the Laplacian polynomial of [Formula: see text]. We finally calculate the Laplacian polynomial of the power graph [Formula: see text].

Graph universal cycles of combinatorial objects

A connected digraph in which the in-degree of any vertex equals its out-degree is Eulerian; this baseline result is used as the basis of existence proofs for universal cycles (also known as ucycles or generalized deBruijn cycles or U-cycles) of several combinatorial objects. The existence of ucycles is often dependent on the specific representation that we use for the combinatorial objects. For example, should we represent the subset {2,5} of {1,2,3,4,5} as “25” in a linear string? Is the representation “52” acceptable? Or is it tactically advantageous (and acceptable) to go with {0,1,0,0,1}? In this paper, we represent combinatorial objects as graphs, as in [3], and exhibit the flexibility and power of this representation to produce graph universal cycles, or Gucycles, for k-subsets of an n-set; permutations (and classes of permutations) of [n]={1,2,…,n}, and partitions of an n-set, thus revisiting the classes first studied in [5]. Under this graphical scheme, we will represent {2,5} as the subgraph A of C5 with edge set consisting of {2,3} and {5,1}, namely the “second” and “fifth” edges in C5. Permutations are represented via their permutation graphs, and set partitions through disjoint unions of complete graphs.

Geodesic orbit and naturally reductive nilmanifolds associated with graphs

AbstractWe study Riemannian nilmanifolds associated with graphs. We prove that such a nilmanifold is geodesic orbit if and only if it is naturally reductive if and only if its defining graph is the disjoint union of complete graphs and the left‐invariant metric is generated by a certain naturally defined inner product.

On [formula omitted]-step competition graphs of bipartite tournaments

In this paper, we study (1,2)-step competition graphs of bipartite tournaments. A bipartite tournament is an orientation of a complete bipartite graph. We show that the (1,2)-step competition graph of a bipartite tournament has at most one non-trivial component or consists of exactly two complete components of size at least three and, especially in the former, the diameter of the nontrivial component is at most three if it exists. Based on this result, we show that, among the connected non-complete graphs which are triangle-free or the cycles of which are edge-disjoint, K1,4 is the only graph that can be represented as the (1,2)-step competition graph of a bipartite tournament. We also completely characterize the complete graphs and the disjoint unions of complete graphs which can be represented as the (1,2)-step competition graph of a bipartite tournament. Finally we present the maximum number of edges and the minimum number of edges which the (1,2)-step competition graph of a bipartite tournament might have.

Distance labellings of Cayley graphs of semigroups

This paper establishes connections between the structure of a semigroup and the minimum spans of distance labellings of its Cayley graphs. We show that certain general restrictions on the minimum spans are equivalent to the semigroup being combinatorial, and that other restrictions are equivalent to the semigroup being a right zero band. We obtain a description of the structure of all semigroups S and their subsets C such that \(\,{\mathrm {Cay}}(S,C)\) is a disjoint union of complete graphs, and show that this description is also equivalent to several restrictions on the minimum span of \(\,{\mathrm {Cay}}(S,C)\). We then describe all graphs with minimum spans satisfying the same restrictions, and give examples to show that a fairly straightforward upper bound for the minimum spans of the underlying undirected graphs of Cayley graphs turns out to be sharp even for the class of combinatorial semigroups.

On Cayley Graphs of some Semigroups

In this paper, if S is a rectangular group, we give some equivalent conditions when Cay ( S,C ) is a disjoint union of complete graphs or P(S) is a D-saturated graph. Also, we investigate the Cayley graphs of completely simple semigroups or completely 0-simple semigroups which are vertex transitive.

Efficient algorithms for cluster editing

The cluster editing problem consists of transforming an input graph $$G$$G into a cluster graph (a disjoint union of complete graphs) by performing a minimum number of edge editing operations. Each edge editing operation consists of either adding a new edge or removing an existing edge. In this paper we propose new theoretical results on data reduction and instance generation for the cluster editing problem, as well as two algorithms based on coupling an exact method to, respectively, a GRASP or ILS heuristic. Experimental results show that the proposed algorithms are able to find high-quality solutions in practical runtime.

Polar permutation graphs are polynomial-time recognisable

Polar graphs generalise bipartite graphs, cobipartite graphs, and split graphs, and they constitute a special type of matrix partitions. A graph is polar if its vertex set can be partitioned into two, such that one part induces a complete multipartite graph and the other part induces a disjoint union of complete graphs. Deciding whether a given arbitrary graph is polar, is an NP-complete problem. Here, we show that for permutation graphs this problem can be solved in polynomial time. The result is surprising, as related problems like achromatic number and cochromatic number are NP-complete on permutation graphs. We give a polynomial-time algorithm for recognising graphs that are both permutation and polar. Prior to our result, polarity has been resolved only for chordal graphs and cographs.

On the existence of total dominating subgraphs with a prescribed additive hereditary property

Recently, Bacsó and Tuza gave a full characterization of the graphs for which every connected induced subgraph has a connected dominating subgraph satisfying an arbitrary prescribed hereditary property. Using their result, we derive a similar characterization of the graphs for which any isolate-free induced subgraph has a total dominating subgraph that satisfies a prescribed additive hereditary property. In particular, we give a characterization for the case where the total dominating subgraphs are a disjoint union of complete graphs. This yields a characterization of the graphs for which every isolate-free induced subgraph has a vertex-dominating induced matching, a so-called induced paired-dominating set.

On the Cayley graphs of completely simple semigroups

In this paper, the Cayley graphs of completely simple semigroups are investigated. The basic structure and properties of this kind of Cayley graph are given, and a condition is given for a Cayley graph of a completely simple semigroup to be a disjoint union of complete graphs. We also describe all pairs (S,A) such that S is a completely simple semigroup, A⊆S, and Cay (S,A) is a strongly connected bipartite Cayley graph.

Disjoint unions of complete graphs characterized by their Laplacian spectrum

A disjoint union of complete graphs is in general not determined by its Laplacian spectrum. It is shown in this paper that if one only considers the family of graphs without isolated vertex, then a disjoint union of complete graphs is determined by its Laplacian spectrum within this family. Moreover, it is shown that the disjoint union of two complete graphs with a and b vertices, a b > 5 3 and b> 1 is determined by its Laplacian spectrum. A counter-example is given when a b = 5 3 .

The Erdős-Ko-Rado properties of various graphs containing singletons

Let G = ( V , E ) be a graph. For r ≥ 1 , let I G ( r ) be the family of independent vertex r -sets of G . For v ∈ V ( G ) , let I G ( r ) ( v ) denote the star { A ∈ I G ( r ) : v ∈ A } . G is said to be r-EKR if there exists v ∈ V ( G ) such that | A | ≤ | I G ( r ) ( v ) | for any non-star family A of pair-wise intersecting sets in I G ( r ) . If the inequality is strict, then G is strictly r - EKR. Let Γ be the family of graphs that are disjoint unions of complete graphs, paths, cycles, including at least one singleton. Holroyd, Spencer and Talbot proved that, if G ∈ Γ and 2 r is no larger than the number of connected components of G , then G is r -EKR. However, Holroyd and Talbot conjectured that, if G is any graph and 2 r is no larger than μ ( G ) , the size of a smallest maximal independent vertex set of G , then G is r -EKR, and strictly so if 2 r < μ ( G ) . We show that in fact, if G ∈ Γ and 2 r is no larger than the independence number of G , then G is r -EKR; we do this by proving the result for all graphs that are in a suitable larger set Γ ′ ⊋ Γ . We also confirm the conjecture for graphs in an even larger set Γ ″ ⊋ Γ ′ .

Representations of disjoint unions of complete graphs

In this paper we will present some new bounds on representation numbers and dimensions of disjoint unions of complete graphs. These representations are closely related to mutually orthogonal sets of latin squares.

On Cayley Graphs of Inverse Semigroups

We describe all finite inverse semigroups and all commutative inverse semigroups with bipartite Cayley graphs. Examples are given which show that this description does not generalize to arbitrary inverse semigroups. Next, we describe all inverse epigroups with Cayley graphs which are disjoint unions of complete graphs. The example of Baer-Levi semigroups shows that it is impossible to drop the condition that G be inverse from this theorem.

Graphs with the Erdős–Ko–Rado property

For a graph G vertex v of G and integer r ⩾ 1 , we denote the family of independent r-sets of V ( G ) by I ( r ) ( G ) and the subfamily { A ∈ I ( r ) ( G ) : v ∈ A } by I v ( r ) ( G ) ; such a subfamily is called a star. Then, G is said to be r-EKR if no intersecting subfamily of I ( r ) ( G ) is larger than the largest star in I ( r ) ( G ) . If every intersecting subfamily of I v ( r ) ( G ) of maximum size is a star, then G is said to be strictly r- EKR. We show that if a graph G is r -EKR then its lexicographic product with any complete graph is r-EKR. For any graph G , we define μ ( G ) to be the minimum size of a maximal independent vertex set. We conjecture that, if 1 ⩽ r ⩽ 1 2 μ ( G ) , then G is r-EKR, and if r < 1 2 μ ( G ) , then G is strictly r-EKR. This is known to be true when G is an empty graph, a cycle, a path or the disjoint union of complete graphs. We show that it is also true when G is the disjoint union of a pair of complete multipartite graphs.

Directed graphs and combinatorial properties of groups and semigroups

This thesis is a study of several combinatorial properties of groups and semigroups. Research on combinatorial properties on groups and semigroups with all infinite subsets having certain unavoidable regularities originates from Ramsey theory and has been the subject of active investigations. In the thesis we explore several properties defined in terms of directed graphs. Chapter 1 outlines some of the earlier achievements in this area and sets the context for the thesis. The thesis continues the work by a number of well-known authors, and contains new results that give rise to interesting connections between graph, group and semigroup theoretic methods. Preliminaries and background information are included in Chapter 2 for convenience of the reader. Several technical facts used in proofs are also referenced. Chapter 3 deals with a combinatorial property related to power graphs. The power graph of a semigroup \(S\) is a directed graph with the set \(S\) of vertices, and with all edges (u, v) such that u ≠ v and v is the power of u. We introduce a combinatorial property defined for power graphs by analogy with several properties considered earlier. The first main theorem completely describes all pairs (\(S\), \(D\)), where \(S\) is a semigroup and \(D\) is a directed graph, and \(S\) satisfies this combinatorial property with respect to D. The structure of the power graphs of all finite abelian groups is then described. Chapter 4 is devoted to Cayley graphs. Let \(T\) be a subset of a semigroup \(S\). The Cayley graph Cay(\({S, T}\)) of \(S\) with respect to \(T\) is defined as the graph with the set \(S\) of vertices and with all edges (x, y), where x ≠ y and xt = y for some t ϵ \(T\). Cayley graphs play important roles in combinatorial group and semigroup theory. For each finite directed graph \(D\), we obtain conditions necessary and sufficient for the Cayley graph Cay(\({S, S})\) of a semigroup \(S\) to contain a subgraph isomorphic to \(D\). The second main theorem of this chapter shows that every infinite semigroup \(S\) has an infinite subset \(T\) inducing a null subgraph in the Cayley graph Cay(\({S, T}\)). A natural question that arises is when the Cayley graph of a semigroup belongs to one of the classes well known in graph theory. The next theorems in this chapter characterise all finite inverse semigroups and all commutative inverse semigroups with bipartite Cayley graphs. We then obtain necessary and sufficient conditions for the Cayley graph of a semigroup to comprise the disjoint union of complete graphs. This result is used to describe all monoids \(S\) and subsets \(T\) of \(S\) such that Cay(\({S, T}\)) is isomorphic to the disjoint union of complete graphs. The divisibility graph of a semigroup \(S\) has edges (u, v), where u belongs to the ideal generated by v. The main theorems of Chapter 5, for each directed graph \(D\), characterise all commutative and completely 0-simple semigroups with all infinite subsets containing divisibility subgraphs isomorphic to \(D\). A description is also given for all monomial matrix semigroups which possess the same property. Chapter 6 examines the concept of an annihilator set. Annihilators have been studied in combinatorial semigroup theory, in particular, in relation to unavoidable regularities in infinite sequences of elements. The annihilator graph of a semigroup \(S\) has edges (u, v) whenever uv = 0 and u ≠ v. The main theorems of this chapter describe all commutative and linear semigroups \(S\), where every infinite subset of \(S\) induces an annihilator graph that has a subgraph isomorphic to a finite directed graph \(D\). The author has also obtained and published new results on other related topics. A few of his theorems on automata, their languages and syntactic monoids have been included in Appendix 1.

Graph Subcolorings: Complexity and Algorithms

In a graph coloring, each color class induces a disjoint union of isolated vertices. A graph subcoloring generalizes this concept, since here each color class induces a disjoint union of complete graphs. Erdos and, independently, Albertson et al., proved that every graph of maximum degree at most 3 has a 2-subcoloring. We point out that this fact is best possible with respect to degree constraints by showing that the problem of recognizing 2-subcolorable graphs with maximum degree 4 is NP-complete, even when restricted to triangle-free planar graphs. Moreover, in general, for fixed k, recognizing k-subcolorable graphs is NP-complete on graphs with maximum degree at most k2 . In contrast, we show that, for arbitrary k, k-SUBCOLORABILITY can be decided in linear time on graphs with bounded treewidth and on graphs with bounded cliquewidth (including cographs as a specific case).

On majority domination in graphs

A majority dominating function on the vertex set of a graph G=( V, E) is a function g: V→{1,−1} such that g( N[ v])⩾1 for at least half of the vertices v in V. The weight of a majority dominating function is denoted as g( V) and is ∑ g(v) over all v in V. The majority domination number of a graph is the minimum possible weight of a majority dominating function, and is denoted as γ maj( G). We determine the majority domination numbers of certain families of graphs. Moreover, we show that the decision problem corresponding to computing the majority domination number of an arbitrary disjoint union of complete graphs is NP-complete.