Drawing Planar Graphs

A dominating set $$D \subseteq V$$D⊆V in a graph $$G=(V,E)$$G=(V,E) is a subset of vertices such that every vertex in V belongs to D or has at least one neighbour in D. In this paper we deal with the problem of finding an approximation of the dominating set of minimum size, i.e., the approximation of the minimum dominating set problem (MDS) in a distributed setting. A distributed algorithm that runs in a constant number of rounds, independent of the size of the network, is called local. In research on distributed local algorithms it is commonly assumed that each vertex has an unique identifier. However, as was shown by Goos et al., for certain classes of graphs (for example, lift-closed bounded degree graphs) identifiers are unnecessary and only a port numbering is needed. We confirm that the same remains true for the MDS up to a constant factor in the class of planar graphs. Namely, we present a local deterministic 694-approximation algorithm for the MDS in planar graphs in a model with a port numbering only. Moreover, our algorithm uses only short messages, i.e., in each round each node can send only a $$O(\log {|V|})$$O(log|V|)-bit message to each of its neighbours.

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

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

On the fixed edge of planar graphs with minimum degree five

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.

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

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.

The geometric simultaneous embedding problem asks whether two planar graphs on the same set of vertices in the plane can be drawn using straight lines, such that each graph is plane. Geometric simultaneous embedding is a current topic in graph drawing and positive and negative results are known for various classes of graphs. So far only connected graphs have been considered. In this paper we present the first results for the setting where one of the graphs is a matching. In particular, we show that there exist a planar graph and a matching which do not admit a geometric simultaneous embedding. This strengthens an analogous negative result for a planar graph and a path. On the positive side, we describe algorithms that compute a geometric simultaneous embedding of a matching and a wheel, outerpath, or tree. Our drawing algorithms minimize the number of orientations used to draw the edges of the matching. Specifically, when embedding a matching and a tree, we can draw all matching edges horizontally. When embedding a matching and a wheel or an outerpath, we use only two orientations.

An octilinear drawing of a planar graph is one in which each edge is drawn as a sequence of horizontal, vertical and diagonal at $45^\circ$ and $-45^\circ$ line-segments. For such drawings to be readable, special care is needed in order to keep the number of bends small. Since the problem of finding planar octilinear drawings with minimum number of bends is NP-hard, in this paper we focus on upper and lower bounds. From a recent result of Keszegh et al. on the slope number of planar graphs, we can derive an upper bound of $4n-10$ bends for planar graphs with $n$ vertices and maximum degree $8$. We considerably improve this general bound and corresponding previous ones for triconnected planar graphs of maximum degree $4$, $5$ and $6$. We also derive non-trivial lower bounds for these three classes of graphs by a technique inspired by the network flow formulation of Tamassia for computing bend optimal orthogonal drawings.

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

Bipartite bithreshold graphs

The notion of word-representable graphs has been extensively studied. It is well known that the set of word-representable graphs are exactly the graphs whose edges can be ordered in a semi-transitive manner. Thus the set of word-representable graphs is decidable. This paper gives an alternative and simpler proof of decidability of word-representable graphs. The second part of the paper introduces a notion called multi-word-representability. Many classes of graphs - planar graphs, interval graphs, split graphs, co-bipartite graphs and line graphs - are shown to be two word-representable. An upper bound on the number of words needed to represent $$k-$$ colourable graphs has also been calculated.

Edge Separators of Planar and Outerplanar Graphs With Applications