Edge Partitions of Optimal 22-plane and 33-plane Graphs

A topological graph is a graph drawn in the plane. A topological graph is $k$-plane, $k>0$, if each edge is crossed at most $k$ times. We study the problem of partitioning the edges of a $k$-plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for $k=1$, we focus on optimal $2$-plane and $3$-plane graphs, which are $2$-plane and $3$-plane graphs with maximum density. We prove the following results. (i) It is not possible to partition the edges of a simple optimal $2$-plane graph into a $1$-plane graph and a forest, while (ii) an edge partition formed by a $1$-plane graph and two plane forests always exists and can be computed in linear time. (iii) We describe efficient algorithms to partition the edges of a simple optimal $2$-plane graph into a $1$-plane graph and a plane graph with maximum vertex degree $12$, or with maximum vertex degree $8$ if the optimal $2$-plane graph is such that its crossing-free edges form a graph with no separating triangles. (iv) We exhibit an infinite family of simple optimal $2$-plane graphs such that in any edge partition composed of a $1$-plane graph and a plane graph, the plane graph has maximum vertex degree at least $6$ and the $1$-plane graph has maximum vertex degree at least $12$. (v) We show that every optimal $3$-plane graph whose crossing-free edges form a biconnected graph can be decomposed, in linear time, into a $2$-plane graph and two plane forests

Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane

Given a set $P$ of $n$ points with their pairwise distances, the traveling salesman problem (TSP) asks for a shortest tour that visits each point exactly once. A TSP instance is rectilinear when the points lie in the plane and the distance considered between two points is the $l_1$ distance. In this paper, a fixed-parameter algorithm for the Rectilinear TSP is presented and relies on techniques for solving TSP on bounded-treewidth graphs. It proves that the problem can be solved in $O\left(nh7^h\right)$ where $h \leq n$ denotes the number of horizontal lines containing the points of $P$. The same technique can be directly applied to the problem of finding a shortest rectilinear Steiner tree that interconnects the points of $P$ providing a $O\left(nh5^h\right)$ time complexity. Both bounds improve over the best time bounds known for these problems.Comment: 24 pages, 13 figures, 6 table