On Balanced Open image in new window-Contact Representations

Abstract

In a Open image in new window -contact representation of a planar graph G, each vertex is represented as an axis-aligned plus shape consisting of two intersecting line segments (or equivalently, four axis-aligned line segments that share a common endpoint), and two plus shapes touch if and only if their corresponding vertices are adjacent in G. Let the four line segments of a plus shape be its arms. In a c-balanced representation, c ≤ 1, every arm can touch at most \(\lceil c\varDelta\rceil\) other arms, where \(\varDelta\) is the maximum degree of G. The widely studied T- and L-contact representations are c-balanced representations, where c could be as large as 1. In contrast, the goal in a c-balanced representation is to minimize c. Let c _k , where k ∈ {2,3}, be the smallest c such that every planar k-tree has a c-balanced representation. In this paper we show that 1/4 ≤ c ₂ ≤ 1/3 ( = b ₂) and 1/3 < c ₃ ≤ 1/2 ( = b ₃). Our result has several consequences. Firstly, planar k-trees admit 1-bend box-orthogonal drawings with boxes of size \(\lceil b_k\varDelta\rceil \times \lceil b_k\varDelta\rceil\), which generalizes a result of Tayu, Nomura, and Ueno. Secondly, they admit 1-bend polyline drawings with \(2\lceil b_k\varDelta\rceil\) slopes, which is significantly smaller than the \(2\varDelta\) upper bound established by Keszegh, Pach, and Pálvölgyi for arbitrary planar graphs.