Complete Multipartite Graphs and their Null Set

For a connected simple graph G , a non-empty set \(S \subseteq V(G)\) of vertices is a safe set if, for every component \(A \text { of }\langle S\rangle_{G}\) and every component \(B \text { of }\langle V(G)-S\rangle_{G}\) adjacent to A , it holds that \(|A| \geq|B|\). The safe number denoted by s(G) of G is the minimum cardinality of a safe set G . In this paper, it examines the characterization of a safe set in complete bipartite graph. It also discusses the minimum cardinality of a safe sets of path graph and cycle graph via modulus. Moreover, this study generates the possible exact values of the safe number of the complete graph, complete bipartite graph, and star graph.

A k-colouring of a graph G with colours 1 , 2 , … , k is canonical with respect to an ordering π = v 1 , v 2 , … , v n of the vertices of G if adjacent vertices are assigned different colours and, for 1 ≤ c ≤ k , whenever colour c is assigned to a vertex v i , each colour less than c has been assigned to a vertex that precedes v i in π . The canonical k-colouring graph of G with respect to π is the graph Can k π ( G ) with vertex set equal to the set of canonical k-colourings of G with respect to π , with two of these being adjacent if and only if they differ in the colour assigned to exactly one vertex. Connectivity and Hamiltonicity of canonical colouring graphs of bipartite and complete multipartite graphs is studied. It is shown that for complete multipartite graphs, and bipartite graphs there exists a vertex ordering π such that Can k π ( G ) is connected for large enough values of k. It is proved that a canonical colouring graph of a complete multipartite graph usually does not have a Hamilton cycle, and that there exists a vertex ordering π such that Can k π ( K m , n ) has a Hamilton path for all k ≥ 3 . The paper concludes with a detailed consideration of Can k π ( K 2 , 2 , … , 2 ) . For each k ≥ χ and all vertex orderings π , it is proved that Can k π ( K 2 , 2 , … , 2 ) is either disconnected or isomorphic to a particular tree.

The Shapley distance in a graph is defined based on Shapley value in cooperative game theory. It is used to measure the cost for a vertex in a graph to access another vertex. In this paper, we establish the Shapley distance between two arbitrary vertices for some special graphs, i.e., path, tree, cycle, complete graph, complete bipartite, and complete multipartite graph. Moreover, based on the Shapley distance, we propose a new index, namely Shapley index, and then compare Shapley index with Wiener index and Kirchhoff index for these special graphs. We also characterize the extremal graphs in which these three indices are equal.

A binary-matroid analogue of a graph connectivity theorem of Jamison and Mulder

In this paper we present some results on the sequence of coefficients of the chromatic polynomial of a graph relative to the complete graph basis, that is, when it is expressed as the sum of the chromatic polynomials of complete graphs. These coefficients are the coefficients of what is often called the σ-polynomial. We obtain necessary and sufficient conditions for this sequence to be symmetrical, and we prove that it is ‘skewed’ and decreasing beyond its midpoint. We also prove that it is strongly log-concave when G is a complete multipartite graph. © 1996 John Wiley & Sons, Inc.

We consider the Gathering problem where a swarm of weak robots disposed on the vertices of an anonymous graph are required to meet at one vertex from where they do not move anymore. In our recent work [Cicerone et al., SIROCCO’19], we have shown how synchronicity heavily affects the design of resolution algorithms within the standard Look-Compute-Move (LCM) model. In particular, we have investigated two dense and highly symmetric topologies: complete graphs and complete bipartite graphs. We characterized all solvable configurations for synchronous robots, whereas it is known that in complete graphs asynchronous robots cannot solve the problem, ever. Instead of approaching directly the asynchronous case in complete bipartite graphs, we asked what happens in the so-called semi-synchronous model, that is robots are synchronized but they are not necessarily all active within all LCM cycles. It turns out that still the gathering can never be accomplished on complete graphs, whereas challenging cases arise in complete bipartite graphs. We provide a distributed algorithm solving the problem for a wide set of possible configurations. For most of the remaining ones instead we provide impossibility results and a few of ad hoc resolution algorithms studied for very specific cases. Over all, still a full characterization is missing but our study points out how difficult might be to derive a general argument that catches all peculiarities. Moreover, some of our approaches reveal new insights that might be very useful for the resolution of other tasks.

The article considers symmetric linear spaces of bipartite graphs (SLSBG), i.e. the set of bipartite graphs with fixed lobes closed with respect to the symmetric difference and permutations of vertices in each lobe. The operation of symmetric difference itself is introduced in this work. The paper provides a structural description of all SLSBG. Symmetric linear spaces of bipartite graphs are divided into trivial (four SLSBG) and nontrivial. Non-trivial ones, in turn, are divided into two families. The first is C -series consisting only of bicomplete graphs, i.e. graphs that are a disjunct union of two complete bipartite graphs graph wings). The second family is D -series that includes graphs in which the degrees of vertices in one lobe have the same parity, and in the other lobe these degrees may be arbitrary. It is proved that every SLSBG of the D -series coincides with one of nine sets defined by the parity of the vertices’ degrees. For the SLSBG of the C -series it is obtained that every two-sided SLSBG (i.e., containing graphs whose both wings have nonempty lobes) is the intersection of the set of all bicomplete graphs with the set of all graphs with an even number of edges or with any space of the D -series.

Lin et al. (2013) [16] conjectured that, the complete multipartite graphs are determined by their distance spectra. This was confirmed by Jin and Zhang (2014) [14]. In the present paper we give a shorter and more readable proof of the conjecture.

We give a unified approach to analyzing, for each positive integer s, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally s‐arc transitive graphs of diameter at least s. A graph is in the class if it is connected and if, for each vertex v, the subgroup of automorphisms fixing v acts transitively on the set of vertices at distance i from v, for each i from 1 to s. We prove that this class is closed under forming normal quotients. Several graphs in the class are designated as degenerate, and a nondegenerate graph in the class is called basic if all its nontrivial normal quotients are degenerate. We prove that, for s≥2, a nondegenerate, nonbasic graph in the class is either a complete multipartite graph or a normal cover of a basic graph. We prove further that, apart from the complete bipartite graphs, each basic graph admits a faithful quasiprimitive action on each of its (1 or 2) vertex‐orbits or a biquasiprimitive action. These results invite detailed additional analysis of the basic graphs using the theory of quasiprimitive permutation groups. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:176‐197, 2012

A graph is a multiclique if its connected components are cliques. A graph is a complete multipartite graph if it is the complement of a multiclique. A graph is a multiclique-multipartite graph if its vertex set has a partition U, W such that G(U) is complete multipartite, G(W) is a multiclique and every two vertices u(U, v(W are adjacent. We describe a polynomial time algorithm to find in polygon-circle graphs a maximum induced complete multipartite subgraph containing an induced K 2,2. In addition, we describe polynomial time algorithms to find maximum induced multicliques and multiclique-multipartite subgraphs in circle graphs. These problems have applications for clustering of proteins by PPI criteria.Keywordspolygon-circle graphcircle graphinduced multicliqueinduced complete multipartite subgraphProtein-Protein-Interaction

Polarity on H-split graphs

In this paper, the lower and upper bounds of the forcing total dr-power dominationnumber of any graph are determined. Total dr-power domination number of some special graphs such as complete graphs, star, fan and wheel graphs are shown. Moreover, the forcing total dr-power domination number of these graphs, together with paths and cycles, are determined.

In this paper we consider the design of systematic low-density parity-check (LDPC) codes using density evolution. We show that systematic LDPC codes can be well represented by tripartite graphs, where the edges in the code graph are divided into two types, source edges and redundancy edges. Using the tripartite code graph representation, we consider the degree distributions for source edges and redundancy edges separately. In particular, we discuss the necessary conditions for the two types of edges to be viewed as a single type in terms of specifying the LDPC codes properly. Then we discuss the relationship between the conventional bipartite graph representation and the tripartite graph representation. For the design of systematic LDPC codes, we show that the tripartite representation can specify the ensembles that are not available for the bipartite representation. Furthermore, we show that codes optimized by deploying the tripartite representation can achieve better performance with respect to the bipartite representation.

For an undirected graph $$G=(V,E)$$G=(V,E), a vertex $$\tau \in V$$źźV separates vertices u and v (where $$u,v\in V$$u,vźV, $$u\ne v$$uźv) if their distances to $$\tau $$ź are not equal. Given an integer parameter $$k \ge 1$$kź1, a set of vertices $$L\subseteq V$$L⊆V is a feasible solution, if for every pair of distinct vertices, u, v, there are at least k distinct vertices $$\tau _{1},\tau _{2},\ldots ,\tau _{k}\in L$$ź1,ź2,ź,źkźL, each separating u and v. Such a feasible solution is called a landmark set, and the k-metric dimension of a graph is the minimal cardinality of a landmark set for the parameter k. The case $$k=1$$k=1 is a classic problem, where in its weighted version, each vertex v has a non-negative cost, and the goal is to find a landmark set with minimal total cost. We generalize the problem for $$k \ge 2$$kź2, introducing two models, and we seek for solutions to both the weighted version and the unweighted version of this more general problem. In the model of all-pairs (AP), k separations are needed for every pair of distinct vertices of V, while in the non-landmarks model (NL), such separations are required only for pairs of distinct vertices in $$V \setminus L$$V\L. We study the weighted and unweighted versions for both models (AP and NL), for path graphs, complete graphs, complete bipartite graphs, and complete wheel graphs, for all values of $$k \ge 2$$kź2. We present algorithms for these cases, thus demonstrating the difference between the two new models, and the differences between the cases $$k=1$$k=1 and $$k \ge 2$$kź2.

We consider the packings and coverings of complete graphs with isomorphic copies of the 4-cycle with a pendant edge. Necessary and sufficient conditions are given for such structures for (1) complete graphs , (2) complete bipartite graphs , and (3) complete graphs with a hole . In the last two cases, we address both restricted and unrestricted coverings.