Spanning Properties of Theta-Theta Graphs
نویسندگان
چکیده
We study the spanning properties of Theta-Theta graphs. Similar in spirit with the Yao-Yao graphs, Theta-Theta graphs partition the space around each vertex into a set of k cones, for some fixed integer k > 1, and select at most one edge per cone. The difference is in the way edges are selected. Yao-Yao graphs select an edge of minimum length, whereas Theta-Theta graphs select an edge of minimum orthogonal projection onto the cone bisector. It has been established that the Yao-Yao graphs with parameter k = 6k′ have spanning ratio 11.67, for k′ ≥ 6. In this paper we establish a first spanning ratio of 7.82 for Theta-Theta graphs, for the same values of k. We also extend the class of Theta-Theta spanners with parameter 6k′, and establish a spanning ratio of 16.76 for k′ ≥ 5. We surmise that these stronger results are mainly due to a tighter analysis in this paper, rather than Theta-Theta being superior to YaoYao as a spanner. We also show that the spanning ratio of Theta-Theta graphs decreases to 4.64 as k′ increases to 8. These are the first results on the spanning properties of Theta-Theta graphs.
منابع مشابه
Theta Graph Designs
We solve the design spectrum problem for all theta graphs with 10, 11, 12, 13, 14 and 15 edges.
متن کاملOn some properties of the hyperspace $theta (X)$ and the study of the space $downarrowtheta C(X)$
The aim of the paper is to first investigate some properties of the hyperspace $theta(X)$, and then in the next article it deals with some detailed study of a special type of subspace $downarrowtheta C(X)$ of the space $theta (Xtimes mathbb I)$.
متن کاملAlgebraic invariants arising from the chromatic polynomials of theta graphs
This paper investigates some algebraic properties of the chromatic polynomials of theta graphs, i.e. graphs which have three internally disjoint paths sharing the same two distinct end vertices. We give a complete description of the Galois group, discriminant and ramification indices for the chromatic polynomials of theta graphs with three consecutive path lengths. We then do the same for theta...
متن کاملUpper Bounds on the Spanning Ratio of Constrained Theta-Graphs
We present tight upper and lower bounds on the spanning ratio of a large family of constrained θ-graphs. We show that constrained θ-graphs with 4k + 2 (k ≥ 1 and integer) cones have a tight spanning ratio of 1 + 2 sin(θ/2), where θ is 2π/(4k+ 2). We also present improved upper bounds on the spanning ratio of the other families of constrained θ-graphs.
متن کاملAre Yao Graph and Theta Graph Void Free?
Greedy Forwarding algorithm is a widely-used routing algorithm for wireless networks. However, it can fail if the wireless network topologies contain void scenarios (or simply called voids). Since Yao Graph and Theta Graph are two types of geometric graphs exploited to construct wireless network topologies, this paper studies whether these two types of graphs can contain voids. Specifically, th...
متن کامل