arXiv subjects

William J. Lenhart

Publications and source records attributed to William J. Lenhart.

5 recordsLinked to original sources

arXiv2023

Mutual Witness Proximity Drawings of Isomorphic Trees

A pair $\langle G_0, G_1 \rangle$ of graphs admits a mutual witness proximity drawing $\langle Γ_0, Γ_1 \rangle$ when: (i) $Γ_i$ represents $G_i$, and (ii) there is an edge $(u,v)$ in $Γ_i$ if and only if there is no vertex $w$ in $Γ_{1-i}$ that is ``too close'' to both $u$ and $v$ ($i=0,1$). In this paper, we consider infinitely many definitions of closeness by adopting the $β$-proximity rule for any $β\in [1,\infty]$ and study pairs of isomorphic trees that admit a mutual witness $β$-proximity drawing. Specifically, we show that every two isomorphic trees admit a mutual witness $β$-proximity drawing for any $β\in [1,\infty]$. The constructive technique can be made ``robust'': For some tree pairs we can suitably prune linearly many leaves from one of the two trees and still retain their mutual witness $β$-proximity drawability. Notably, in the special case of isomorphic caterpillars and $β=1$, we construct linearly separable mutual witness Gabriel drawings.

cs.CG↗

arXiv2022

On the Complexity of the Storyplan Problem

Motivated by dynamic graph visualization, we study the problem of representing a graph $G$ in the form of a \emph{storyplan}, that is, a sequence of frames with the following properties. Each frame is a planar drawing of the subgraph of $G$ induced by a suitably defined subset of its vertices. Between two consecutive frames, a new vertex appears while some other vertices may disappear, namely those whose incident edges have already been drawn in at least one frame. In a storyplan, each vertex appears and disappears exactly once. For a vertex (edge) visible in a sequence of consecutive frames, the point (curve) representing it does not change throughout the sequence. Note that the order in which the vertices of $G$ appear in the sequence of frames is a total order. In the \textsc{StoryPlan} problem, we are given a graph and we want to decide whether there exists a total order of its vertices for which a storyplan exists. We prove that the problem is NP-complete, and complement this hardness with two parameterized algorithms, one in the vertex cover number and one in the feedback edge set number of $G$. Also, we prove that partial $3$-trees always admit a storyplan, which can be computed in linear time. Finally, we show that the problem remains NP-complete in the case in which the total order of the vertices is given as part of the input and we have to choose how to draw the frames.

cs.CC↗

arXiv2022

Mutual Witness Gabriel Drawings of Complete Bipartite Graphs

Let $Γ$ be a straight-line drawing of a graph and let $u$ and $v$ be two vertices of $Γ$. The Gabriel disk of $u,v$ is the disk having $u$ and $v$ as antipodal points. A pair $\langle Γ_0,Γ_1 \rangle$ of vertex-disjoint straight-line drawings form a mutual witness Gabriel drawing when, for $i=0,1$, any two vertices $u$ and $v$ of $Γ_i$ are adjacent if and only if their Gabriel disk does not contain any vertex of $Γ_{1-i}$. We characterize the pairs $\langle G_0,G_1 \rangle $ of complete bipartite graphs that admit a mutual witness Gabriel drawing. The characterization leads to a linear time testing algorithm. We also show that when at least one of the graphs in the pair $\langle G_0, G_1 \rangle $ is complete $k$-partite with $k>2$ and all partition sets in the two graphs have size greater than one, the pair does not admit a mutual witness Gabriel drawing.

cs.CG↗

arXiv2018

(k,p)-Planarity: A Relaxation of Hybrid Planarity

We present a new model for hybrid planarity that relaxes existing hybrid representations. A graph $G = (V,E)$ is $(k,p)$-planar if $V$ can be partitioned into clusters of size at most $k$ such that $G$ admits a drawing where: (i) each cluster is associated with a closed, bounded planar region, called a cluster region; (ii) cluster regions are pairwise disjoint, (iii) each vertex $v \in V$ is identified with at most $p$ distinct points, called \emph{ports}, on the boundary of its cluster region; (iv) each inter-cluster edge $(u,v) \in E$ is identified with a Jordan arc connecting a port of $u$ to a port of $v$; (v) inter-cluster edges do not cross or intersect cluster regions except at their endpoints. We first tightly bound the number of edges in a $(k,p)$-planar graph with $p<k$. We then prove that $(4,1)$-planarity testing and $(2,2)$-planarity testing are NP-complete problems. Finally, we prove that neither the class of $(2,2)$-planar graphs nor the class of $1$-planar graphs contains the other, indicating that the $(k,p)$-planar graphs are a large and novel class.

cs.DS↗

arXiv2016

On Partitioning the Edges of 1-Plane Graphs

A 1-plane graph is a graph embedded in the plane such that each edge is crossed at most once. A 1-plane graph is optimal if it has maximum edge density. A red-blue edge coloring of an optimal 1-plane graph $G$ partitions the edge set of $G$ into blue edges and red edges such that no two blue edges cross each other and no two red edges cross each other. We prove the following: $(i)$ Every optimal 1-plane graph has a red-blue edge coloring such that the blue subgraph is maximal planar while the red subgraph has vertex degree at most four; this bound on the vertex degree is worst-case optimal. $(ii)$ A red-blue edge coloring may not always induce a red forest of bounded vertex degree. Applications of these results to graph augmentation and graph drawing are also discussed.

cs.CG↗