Edge-Constrained Hamiltonian Paths on a Point Set

TL;DR

This work investigates edge-constrained plane Hamiltonian paths on point sets S in general position. It completely characterizes the existence and provides constructions for a single s–t path under constraints that a fixed segment ab is either included or avoided, and extends the analysis to two plane Hamiltonian paths under similar segment constraints, including cases of edge-disjointness and shared segments. Central tools include obstructions such as convex-hull configurations and wheel-like scenarios, together with algorithmic frameworks that achieve $O(n \log n)$ time for constructions, aided by crossing-dominance concepts and the AB-Alternating path lemma. The results advance the understanding of geometric graph packing with constraints and pave the way for more general edge- and endpoint-prescribing reconfiguration problems in plane graphs.

Abstract

Let S be a set of distinct points in general position in the Euclidean plane. A plane Hamiltonian path on S is a crossing-free geometric path such that every point of S is a vertex of the path. It is known that, if S is sufficiently large, there exist three edge-disjoint plane Hamiltonian paths on S. In this paper we study an edge-constrained version of the problem of finding Hamiltonian paths on a point set. We first consider the problem of finding a single plane Hamiltonian path pi with endpoints s, t in S and constraints given by a segment ab, where a, b in S. We consider the following scenarios: (i) ab in pi; (ii) ab not in pi. We characterize those quintuples (S, a, b, s, t) for which pi exists. Secondly, we consider the problem of finding two plane Hamiltonian paths pi_1, pi_2 on a set S with constraints given by a segment ab, where a, b in S. We consider the following scenarios: (i) pi_1 and pi_2 share no edges and ab is an edge of pi_1; (ii) pi_1 and pi_2 share no edges and none of them includes ab as an edge; (iii) both pi_1 and pi_2 include ab as an edge and share no other edges. In all cases, we characterize those triples (S, a, b) for which pi_1 and pi_2 exist.

Figures (14)

• Figure 1: Two Hamiltonian paths on $4$ points with neighboring starting points $s,t$.
• Figure 2: Sets satisfying the conditions of \ref{['lem:convexobstacle']} (left) and \ref{['lem:wheelobstacle']} (right).
• Figure 3: Construction of path $\pi$ in \ref{['thm:stpathsegnotinc']}.
• Figure 4: Sets satisfying the conditions of \ref{['lem:bottomobstruction']} (left) and \ref{['lem:complicatedobstruction']} (right), with possible bridges drawn in dashed blue.
• Figure 6: Construction of $\pi_1$ and $\pi_2$ when both $a$ and $b$ are vertices of $\partial \mathop{\mathrm{CH}}\nolimits(S)$.

Theorems & Definitions (53)

• lemma thmcounterlemma: KindermannKLV23, Lemma 5
• lemma thmcounterlemma: KindermannKLV23, Theorem 2
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma: lem:pathGivenSegment
• lemma thmcounterlemma: lem:convexobstacle
• lemma thmcounterlemma: lem:wheelobstacle
• theorem thmcountertheorem: thm:stpathsegnotinc
• lemma thmcounterlemma: lem:oneendptsegnotinc
• proof : of \ref{['thm:stpathsegnotinc']}