Multithreshold multipartite graphs with small parts

A graph Γ is called a Deza graph if it is regular and the number of common neighbors of any two distinct vertices is one of two fixed values. A Deza graph is called a strictly Deza graph if it has diameter 2 and is not strongly regular. In 1992, Gardiner et al. proved that a strongly regular graph that contains a vertex with disconnected second neighborhood is a complete multipartite graph with parts of the same size greater than 2. In this paper, we study strictly Deza graphs with disconnected second neighborhoods of vertices. In Section 2, we prove that, if each vertex of a strictly Deza graph has disconnected second neighborhood, then the graph is either edge-regular or coedge-regular. In Sections 3 and 4, we consider strictly Deza graphs that contain at least one vertex with disconnected second neighborhood. In Section 3, we show that, if such a graph is edge-regular, then it is the s-coclique extension of a strongly regular graph with parameters (n, k, λ, μ), where s is an integer, s ≥ 2, and λ = μ. In Section 4, we show that, if such a graph is coedge-regular, then it is the 2-clique extension of a complete multipartite graph with parts of the same size greater than or equal to 3.

The Study of algebraic structures, especially on graphs theory, leads to anew topics of research in recent years. In this paper, the algebraic structures that will be represented by a coprime graph are the dihedral group and its subgroups. The coprime graph of a group G, denoted by \Gamma_D_2n is a graph whose vertices are elements of G and two distinct vertices a and b are adjacent if only if (|a,|b|)=1. Some properties of the coprime graph of a dihedral group D_2n are obtained. One of the results is if n is prime then \Gamma_D_2n is a complete bipartite graph. Moreover, if n is the power of prime then \Gamma_D_2n is a multipartite graph.

Peer-to-peer energy trading in energy local area network considering decentralized energy routing

The vertex arboricity va(G) of a graph G is the minimum number of subsets into which the vertex set V(G) can be partitioned so that each subset induces an acyclic subgraph. The fractional version of vertex arboricity is introduced in this paper. We determine fractional vertex arboricity for several classes of graphs, e.g., complete multipartite graphs, cycles, integer distance graphs, prisms and Peterson graph.

In this paper we consider a colouring version of the general position problem. The gp-chromatic number is the smallest number of colours needed to colour V(G) such that each colour class has the no-three-in-line property. We determine bounds on this colouring number in terms of the diameter, general position number, size, chromatic number, cochromatic number and total domination number and prove realisation results. We also determine it for several graph classes, including Kneser graphs K(n,2), line graphs of complete graphs, complete multipartite graphs, block graphs and Cartesian products. We also show that the gp-colouring problem is NP-complete.

Let $G$ be a simple connected graph and $\mu_1(G) \geq \mu_2(G) \geq \cdots \geq \mu_n(G)$ be the Laplacian eigenvalues of $G$. Let $\overline{G}$ be the complement of $G$. Einollahzadeh et al.[J. Combin. Theory Ser. B, 151(2021), 235–249] proved that $\mu_{n-1}(G)+\mu_{n-1}(\overline{G})\geq 1$. Grijò et al. [Discrete Appl. Math., 267(2019), 176–183] conjectured that $\mu_{n-2}(G)+\mu_{n-2}(\overline{G})\geq 2$ for any graph and proved it to be true for some graphs. In this paper, we prove $\mu_{n-2}(G)+\mu_{n-2}(\overline{G})\geq 2$ is true for some new graphs. Furthermore, we propose a more general conjecture that $\mu_k(G)+\mu_k(\overline{G})\geq n-k$ holds for any graph $G$, with equality if and only if $G$ or $\overline{G}$ is isomorphic to $K_{n-k}\vee H$, where $H$ is a disconnected graph on $k$ vertices and has at least $n-k+1$ connected components. And we prove that it is true for $k\leq \frac{n+1}{2}$, for unicyclic graphs, bicyclic graphs, threshold graphs, bipartite graphs, regular graphs, complete multipartite graphs and c-cyclic graphs when $n\geq 2c+8$.

It is a long standing open problem to find an explicit description of the stable set polytope of claw-free graphs. Yet more than 20 years after the discovery of a polynomial algorithm for the maximum stable set problem for claw-free graphs, there is even no conjecture at hand today.Such a conjecture exists for the class of quasi-line graphs. This class of graphs is a proper superclass of line graphs and a proper subclass of claw-free graphs for which it is known that not all facets have 0/1 normal vectors. Ben Rebea’s conjecture states that the stable set polytope of a quasi-line graph is completely described by clique-family inequalities. Chudnovsky and Seymour recently provided a decomposition result for claw-free graphs and proved that Ben Rebea’s conjecture holds, if the quasi-line graph is not a fuzzy circular interval graph.In this paper, we give a proof of Ben Rebea’s conjecture by showing that it also holds for fuzzy circular interval graphs. Our result builds upon an algorithm of Bartholdi, Orlin and Ratliff which is concerned with integer programs defined by circular ones matrices.KeywordsNegative CycleClique InequalityHeavy EdgeRational PolyhedronLight EdgeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The notion of a boundary graph class was recently introduced for a classification of hereditary graph classes according to the complexity of a considered problem. Two concrete graph classes are known to be boundary for several graph problems. We formulate a criterion to determine whether these classes are boundary for a given graph problem or not. We also demonstrate that the classes are simultaneously boundary for some continuous set of graph problems and they are not simultaneously boundary for another set of the same cardinality. Both families of problems are constituted by variants of the maximum induced subgraph problem.

We study various optimization problems in t-subtree graphs, the intersection graphs of t-subtrees, where a t-subtree is the union of t disjoint subtrees of some tree. This graph class generalizes both the class of chordal graphs and the class of t-interval graphs, a generalization of interval graphs that has recently been studied from a combinatorial optimization point of view. We present approximation algorithms for the Maximum Independent Set, Minimum Coloring, Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique problems in t-subtree graphs.

Polarity on H-split graphs

We present an effective heuristic for the Steiner Problem in Graphs. Its main elements are a multistart algorithm coupled with aggressive combination of elite solutions, both leveraging recently-proposed fast local searches. We also propose a fast implementation of a well-known dual ascent algorithm that not only makes our heuristics more robust (by dealing with easier cases quickly), but can also be used as a building block of an exact branch-and-bound algorithm that is quite effective for some inputs. On all graph classes we consider, our heuristic is competitive with (and sometimes more effective than) any previous approach with similar running times. It is also scalable: with long runs, we could improve or match the best published results for most open instances in the literature.

A graph is called Hamiltonian connected if it contains a Hamiltonian path between any two distinct vertices. In the past, we proved the Hamiltonian path and cycle problems for general supergrid graphs to be NP-complete. However, they are still open for solid supergrid graphs. In this paper, first we will verify the Hamiltonian cycle property of C-shaped supergrid graphs, which are a special case of solid supergrid graphs. Next, we show that C-shaped supergrid graphs are Hamiltonian connected except in a few conditions. For these excluding conditions of Hamiltonian connectivity, we compute their longest paths. Then, we design a linear-time algorithm to solve the longest path problem in these graphs. The Hamiltonian connectivity of C-shaped supergrid graphs can be applied to compute the optimal stitching trace of computer embroidery machines, and construct the minimum printing trace of 3D printers with a C-like component being printed.

Matching interdiction

Let $G$ be an $r$-partite graph such that the edge density between any two parts is at least $\alpha$. How large does $\alpha$ need to be to guarantee that $G$ contains a connected transversal, that is, a tree on $r$ vertices meeting each part in one vertex? And what if instead we want to guarantee the existence of a Hamiltonian transversal? In this paper we initiate the study of such extremal multipartite graph problems, obtaining a number of results and providing many new constructions, conjectures and further questions.

For any positive integer k, the total k-domatic partition problem is to partition the vertices of a graph G into k pairwise disjoint total dominating sets. In this paper, we study the problem for planar graphs, chordal bipartite graphs, convex bipartite graphs, and bipartite permutation graphs. We show that the total 3-domatic partition problem on planar graphs is NP-complete. Moreover, we give an alternative algorithm to solve the total k-domatic partition problem for chordal bipartite graphs with weak elimination orderings, and adapt it to solve the problem in linear time for bipartite permutation graphs and convex bipartite graphs even if Gamma-free forms of the adjacency matrices of the considered graphs are not given.