Total domination in plane triangulations

Preprin

Extremal Problems For Transversals In Graphs With Bounded Degree

We introduce and discuss generalizations of the problem of independent transversals. Given a graph property {\user1{\mathcal{R}}} , we investigate whether any graph of maximum degree at most d with a vertex partition into classes of size at least p admits a transversal having property {\user1{\mathcal{R}}} . In this paper we study this problem for the following properties {\user1{\mathcal{R}}} : "acyclic”, "H-free”, and "having connected components of order at most r”. We strengthen a result of [13]. We prove that if the vertex set of a d-regular graph is partitioned into classes of size d+⌞d/r⌟, then it is possible to select a transversal inducing vertex disjoint trees on at most r vertices. Our approach applies appropriate triangulations of the simplex and Sperner's Lemma. We also establish some limitations on the power of this topological method. We give constructions of vertex-partitioned graphs admitting no independent transversals that partially settles an old question of Bollobás, Erdős and Szemerédi. An extension of this construction provides vertex-partitioned graphs with small degree such that every transversal contains a fixed graph H as a subgraph. Finally, we pose several open question

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Positivity of the universal pairing in 3 dimensions

Associated to a closed, oriented surface S is the complex vector space with basis the set of all compact, oriented 3-manifolds which it bounds. Gluing along S defines a Hermitian pairing on this space with values in the complex vector space with basis all closed, oriented 3-manifolds. The main result in this paper is that this pairing is positive, i.e. that the result of pairing a nonzero vector with itself is nonzero. This has bearing on the question of what kinds of topological information can be extracted in principle from unitary 2+1 dimensional TQFTs. The proof involves the construction of a suitable complexity function c on all closed 3-manifolds, satisfying a gluing axiom which we call the topological Cauchy-Schwarz inequality, namely that c(AB) <= max(c(AA),c(BB)) for all A,B which bound S, with equality if and only if A=B. The complexity function c involves input from many aspects of 3-manifold topology, and in the process of establishing its key properties we obtain a number of results of independent interest. For example, we show that when two finite volume hyperbolic 3-manifolds are glued along an incompressible acylindrical surface, the resulting hyperbolic 3-manifold has minimal volume only when the gluing can be done along a totally geodesic surface; this generalizes a similar theorem for closed hyperbolic 3-manifolds due to Agol-Storm-Thurston.Comment: 83 pages, 21 figures; version 3: incorporates referee's comments and correction

Efficient Data Dissemination in Wireless Ad Hoc Networks

In this thesis, we study the problem of efficient data dissemination in wireless sensor and mobile ad hoc networks. In wireless sensor networks we study two problems: (1) construction of virtual backbones and clustering hierarchies to achieve efficient routing, and (2) placement of multiple sinks, where each sensor is at a bounded distance to several sinks, to analyze and process data before sending it to a central unit. Often connected dominating sets have been used for such purposes. However, a connected dominating set is often vulnerable due to frequent node failures in wireless sensor networks. Hence, to provide a degree of fault-tolerance we consider in problem (1) a 2-connected (k,r)-dominating set, denoted D(2,k,r), to act as a virtual backbone or a clustering hierarchy, and in problem (2) a total (k,r)-dominating set to act as sinks in wireless sensor networks. Ideally, the backbone or the number of sinks in the network should constitute the smallest percentage of nodes in the network. We model the wireless sensor network as a graph. The total (k,r)-dominating set and the 2-connected (k,r)-dominating set have not been studied in the literature. Thus, we propose two centralized approximation algorithms to construct a D(2,k,r) in unit disk graphs and in general graphs. We also derive upper bounds on the total (k,r)-domination number in graphs of girth at least 2k+1 as well as in random graphs with non-fixed probability p. In mobile ad hoc networks we propose a hexagonal based beacon-less flooding algorithm, HBLF, to efficiently flood the network. We give sufficient condition that even in the presence of holes in the network, HBLF achieves full delivery. Lower and upper bounds are given on the number of forwarding nodes returned by HBLF in a network with or without holes. When there are no holes in the network, we show that the ratio of the shortest path returned by HBLF to the shortest path in the network is constant. We also present upper bounds on the broadcast time of HBLF in a network with or without holes