A local approximation algorithm for minimum dominating set problem in anonymous planar networks

In this paper we consider a generalization of the classical dominating set problem to the k-tuple dominating set problem (kMDS). For any positive integer k, we look for a smallest subset of vertices D ⊆ V with the property that every vertex in V ∖ D is adjacent to at least k vertices of D. We are interested in the distributed complexity of this problem in the model, where the nodes have no identifiers. The most challenging case is when k = 2, and for this case we propose a distributed local algorithm, which runs in a constant number of rounds, yielding a 7-approximation in the class of planar graphs. On the other hand, in the class of algorithms in which every vertex uses only its degree and the degree of its neighbors to make decisions, there is no algorithm providing a (5 − e)-approximation of the 2MDS problem. In addition, we show a lower bound of (4 − e) for the 2MDS problem even if unique identifiers are allowed.

When we want to draw a graph to make the information contained in its structure easily accessible, it is highly desirable to have a drawing with as few edge crossings as possible (Purchase et al., 1997; Purchase, 1997). The class of graphs that can be drawn with no crossings at all is the class of planar graphs. Algorithms for drawing planar graphs are the main subject of this chapter.

Improved distributed local approximation algorithm for minimum 2-dominating set in planar graphs

In this paper, we study the class of graphs Gm,n that have the same degree sequence as two disjoint cliques Km and Kn, as well as the class G¯m,n of the complements of such graphs. The problems of finding a maximum clique and a maximum independent set are NP-hard on Gm,n. Therefore, looking for upper and lower bounds for the clique and independence numbers of such graphs is a challenging task. In this article, we obtain such bounds, as well as other related results. In particular, we consider the class of regular graphs, which are degree-equivalent to arbitrarily many identical cliques, as well as such graphs of bounded degree.

For a connected graph $$G = \left( V,E\right) $$ , a set $$S\subseteq E(G)$$ is called a total edge-to-vertex monophonic set of a connected graph G if the subgraph induced by S has no isolated edges. The total edge-to-vertex monophonic number $$m_{tev}(G)$$ of G is the minimum cardinality of its total edge-to-vertex monophonic set of G. The total edge-to-vertex monophonic number of certain classes of graphs is determined and some of its general properties are studied. Connected graphs of size $$q \ge 3 $$ with total edge-to-vertex monophonic number q is characterized. It is shown that for positive integers $$r_{m},d_{m}$$ and $$l\ge 4$$ with $$r_{m}< d_{m} \le 2 r_{m}$$ , there exists a connected graph G with $$\textit{rad}_ {m} G = r_{m}$$ , $$\textit{diam}_ {m} G = d_{m}$$ and $$m_{tev}(G) = l$$ and also shown that for every integers a and b with $$2 \le a \le b$$ , there exists a connected graph G such that $$ m_{ev}\left( G\right) = b$$ and $$m_{tev}(G) = a + b$$ . A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing total edge-to-vertex monophonic number of S, denoted by $$f_{tev}(S)$$ is the cardinality of a minimum forcing subset of S. The forcing total edge-to-vertex monophonic number of G, denoted by $$f_{tev}(G) = \textit{min}\{f_{tev}(S)\}$$ , where the minimum is taken over all total edge-to-vertex monophonic set S in G. The forcing total edge-to-vertex monophonic number of certain classes of graphs are determined and some of its general properties are studied. It is shown that for every integers a and b with $$0 \le a \le b$$ and $$b \ge 2$$ , there exists a connected graph G such that $$f_{tev}(G) = a$$ and $$ m _{tev}(G) = b$$ , where $$ f _{tev}(G)$$ is the forcing total edge-to-vertex monophonic number of G.

This study investigates the (a,d)-face antimagic labeling on three classes of planar graphs, namely the house graph, the double house graph, and the triangular ladder graph. An (a,d)-face antimagic labeling is defined as a bijective function from the set of vertices, edges, and faces to the set of positive integers, such that the sum of the labels assigned to the vertices and edges incident with each face forms an arithmetic sequence. In this work, specific labeling constructions are developed for the vertices, edges, and faces of each graph to ensure that the face antimagic property is satisfied. Illustrative examples are presented to demonstrate the labeling schemes that produce face weights forming an arithmetic sequence with a fixed common difference of d = 1. The findings confirm that all three classes of graphs admit valid face antimagic labelings. This research contributes to the advancement of graph labeling theory, particularly in the study of structured planar graphs

On the hull number on cycle convexity of graphs

Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances

We prove that the distance-r dominating set, distance-r connected dominating set, distance-r vertex cover, and distance-r connected vertex cover problems admit constant factor approximations in the CONGEST model of distributed computing in a constant number of rounds on classes of sparse high-girth graphs. In this paper, sparse means bounded expansion, and high-girth means girth at least 4r+2. Our algorithm is quite simple; however, the proof of its approximation guarantee is non-trivial. To complement the algorithmic results, we show tightness of our approximation by providing a loosely matching lower bound on rings.Our result is the first to show the existence of constant-factor approximations in a constant number of rounds in non-trivial classes of graphs for distance-r covering problems.

On the fixed edge of planar graphs with minimum degree five

Journal Article Planar and Quasi-Planar Simultaneous Geometric Embedding Get access Emilio Di Giacomo, Emilio Di Giacomo * 1Dipartimento di Ingegneria, Università degli Studi di Perugia, Perugia, Italy *Corresponding author: emilio.digiacomo@unipg.it Search for other works by this author on: Oxford Academic Google Scholar Walter Didimo, Walter Didimo 1Dipartimento di Ingegneria, Università degli Studi di Perugia, Perugia, Italy Search for other works by this author on: Oxford Academic Google Scholar Giuseppe Liotta, Giuseppe Liotta 1Dipartimento di Ingegneria, Università degli Studi di Perugia, Perugia, Italy Search for other works by this author on: Oxford Academic Google Scholar Henk Meijer, Henk Meijer 2University College Roosevelt, Middelburg, The Netherlands Search for other works by this author on: Oxford Academic Google Scholar Stephen K. Wismath Stephen K. Wismath 3Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Canada Search for other works by this author on: Oxford Academic Google Scholar The Computer Journal, Volume 58, Issue 11, November 2015, Pages 3126–3140, https://doi.org/10.1093/comjnl/bxv048 Published: 13 July 2015 Article history Received: 22 October 2014 Revision received: 22 May 2015 Published: 13 July 2015

An embedding of a graph in a book consists of a linear order of its vertices along the spine of the book and of an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. Accordingly, the book thickness of a class of graphs is the maximum book thickness over all its members. In this paper, we address a long-standing open problem regarding the exact book thickness of the class of planar graphs, which previously was known to be either three or four. We settle this problem by constructing planar graphs that require four pages in all of their book embeddings, thus establishing that the book thickness of the class of planar graphs is four.

Baker [1] devised a technique to obtain approximation schemes for many optimization problems restricted to planar graphs; her technique was later extended to more general graph classes. In particular, using the Baker's technique and the minor structure theorem, Dawar et al. [5] gave Polynomial-Time Approximation Schemes (PTAS) for all monotone optimization problems expressible in the first-order logic when restricted to a proper minor-closed class of graphs. We define a Baker game formalizing the notion of repeated application of Baker's technique interspersed with vertex removal, prove that monotone optimization problems expressible in the first-order logic admit PTAS when restricted to graph classes in which the Baker game can be won in a constant number of rounds, and prove without use of the minor structure theorem that all proper minor-closed classes of graphs have this property.

Homomorphisms of partial t-trees and edge-colorings of partial 3-trees

Let [Formula: see text] be a finite and simple undirected connected graph of order [Formula: see text] and let [Formula: see text] be a proper vertex coloring of [Formula: see text]. Denote [Formula: see text] simply, [Formula: see text]. In this paper, we introduce a variation of the well-known Zagreb indices by utilizing the parameter [Formula: see text] instead of the invariant [Formula: see text] for all vertices of [Formula: see text]. The new indices are called chromatic Zagreb indices. We study these new indices for certain classes of graphs and introduce the notion of chromatically stable graphs.