Upward Pointset Embeddings of Planar st-Graphs

TL;DR

UPSE Testing for planar $st$-graphs is NP-hard, even for graphs formed by internally-disjoint $st$-paths and for directed root-to-leaf trees, via a 3-Partition reduction; the problem becomes tractable when the maximum $st$-cutset size $k$ is bounded, solvable in $O(n^{4k})$ time with $O(n^{3k})$ space, and all UPSEs can be enumerated with $O(n)$ delay after a $O(k n^{4k} \log n)$ setup. The authors develop a dynamic-programming framework using $st$-cutset–to–horizontal-line correspondences, enabling both testing and enumeration, and they extend the approach to two-paths graphs, yielding an $O(n \log n)$ test in general position. They also connect UPSEs to non-crossing monotone Hamiltonian cycles on the pointset and provide an $O(n)$-delay enumeration algorithm for these cycles with $O(n^2)$ setup/space, offering practical polygonalization insights. Overall, the work advances the theory and algorithms for upward pointset embeddings, identifying hardness frontiers and efficient, structured algorithms for key graph families and for polygonalizations tied to UPSEs.

Abstract

We study upward pointset embeddings (UPSEs) of planar $st$-graphs. Let $G$ be a planar $st$-graph and let $S \subset \mathbb{R}^2$ be a pointset with $|S|= |V(G)|$. An UPSE of $G$ on $S$ is an upward planar straight-line drawing of $G$ that maps the vertices of $G$ to the points of $S$. We consider both the problem of testing the existence of an UPSE of $G$ on $S$ (UPSE Testing) and the problem of enumerating all UPSEs of $G$ on $S$. We prove that UPSE Testing is NP-complete even for $st$-graphs that consist of a set of directed $st$-paths sharing only $s$ and $t$. On the other hand, if $G$ is an $n$-vertex planar $st$-graph whose maximum $st$-cutset has size $k$, then UPSE Testing can be solved in $O(n^{4k})$ time with $O(n^{3k})$ space; also, all the UPSEs of $G$ on $S$ can be enumerated with $O(n)$ worst-case delay, using $O(k n^{4k} \log n)$ space, after $O(k n^{4k} \log n)$ set-up time. Moreover, for an $n$-vertex $st$-graph whose underlying graph is a cycle, we provide a necessary and sufficient condition for the existence of an UPSE on a given pointset, which can be tested in $O(n \log n)$ time. Related to this result, we give an algorithm that, for a set $S$ of $n$ points, enumerates all the non-crossing monotone Hamiltonian cycles on $S$ with $O(n)$ worst-case delay, using $O(n^2)$ space, after $O(n^2)$ set-up time.

Figures (11)

• Figure 1: Illustration for the proof of \ref{['th:st-hardness']}. (a) The pointset $S$. (b) The UPSE of $G$ on $S$, where the $a_i$-paths are drawn in red and the additional $k$-paths are in blue. The pointset $S$ and the graph $G$ are those resulting from the reduction applied to the instance $A=\{2,2,2,2,2,2,3,3,3,3,4,4\}$.
• Figure 2: Illustration for the proof of \ref{['th:tree-hardness']}. (a) The pointset $S$. The points of $S$ visible from $p_1$ (green points) are as many as the children of the root of the tree $T$. The portions of the regions $R_1,R_2,\dots,R_b$ below the line $\ell$ are alternately colored gray and white. (b) The UPSE of $T$ on $S$ corresponding to a solution to the original instance 3-partition (red vertices).
• Figure 3: (a) An entry $\chi = \bigcup^5_{i=1} \langle e_i,p_i,q_i \rangle$ with $T[\space{\chi}\space] = \texttt{True}$ and a corresponding UPSE of $G_\chi$ on a subset of $S$ that includes $p_s$. The edges in $E(\chi)$ are colored blue. (b) An entry $\varphi$ from which $\chi$ stems; the points in $S_\downarrow$ are filled white. The edges in $H^-$ are colored green, while the edges in $H^+$ are colored orange.
• Figure 4: Illustrations for \ref{['cl:st-cutset']}. (a) The connected components $C_s$ (dashed) and $C_t$ (solid black) defined by the $st$-cuteset $E(\chi)$. (b) The connected components $C'_s$ (dashed) and $C'_t$ (solid black) defined by the $st$-cuteset $H$ (blue and orange edges).
• Figure 5: Illustration for the base case of \ref{['th:two-paths']}, when $S_L = \{p_s, p_t\}$ and $|\mathop{\mathrm{\mathcal{H}_R}}\nolimits(S)| = |P_R|$. The drawing of $P_R$ coincides with $\mathcal{E}_R(S)$.

Theorems & Definitions (17)

• theorem 1.1
• proof
• theorem 1.2
• proof
• Claim 1.3
• proof
• Claim 1.5
• proof
• Claim 1.6
• proof