On RAC Drawings of Graphs with Two Bends per Edge

Abstract

It is shown that every $n$-vertex graph that admits a 2-bend RAC drawing in the plane, where the edges are polylines with two bends per edge and any pair of edges can only cross at a right angle, has at most $20n-24$ edges for $n\geq 3$. This improves upon the previous upper bound of $74.2n$; this is the first improvement in more than 12 years. A crucial ingredient of the proof is an upper bound on the size of plane multigraphs with polyline edges in which the first and last segments are either parallel or orthogonal.

Figures (6)

• Figure 1: (a) A plane ortho-fin multigraph with 3 vertices and 7 edges. The parallel edges $e$ and $f$ form a simple polygon $P$ with potential $\Phi(P)=\pi/2$. (b) A plane ortho-fin multigraph with 2 vertices and 7 edges.
• Figure 2: A plane ortho-fin multigraph $G$ with a cut vertex $v_0$; and its decomposition into three plane ortho-fin multigraphs $G_1$, $G_2$ and $G_3$. Face $P$ is the intersection of a bounded face $P_1$ of $G_1$ and two unbounded faces $P_2$ and $P_3$ of $G_2$ and $G_3$, respectively.
• Figure 3: Construction for plane ortho-fin multigraphs with $n=2,3,4$ vertices and $5n-2$ edges.
• Figure 4: (a) Three blocks in a RAC$_2$ drawing. (b) A spanning tree $T$, after splitting five terminals in $V$ into nine terminals (red dots). (c) A plane ortho-fin multigraph on the five terminals in $V$.
• Figure 5: Graphs $H$ (left), $G_1$ (middle) and $G_2$ (right) for the RAC$_2$ drawing in Fig. \ref{['fig:2']}. The original RAC$_2$ drawing is shown in light gray for comparison.

Theorems & Definitions (11)

• Theorem 1
• Lemma 2
• proof
• Theorem 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6