Reconfiguration of Hamiltonian cycles and paths in grid graphs
Date
2020-05-11
Authors
Nishat, Rahnuma Islam
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Abstract
A grid graph is a finite embedded subgraph of the infinite integer grid. A solid
grid graph is a grid graph without holes, i.e., each bounded face of the graph is a
unit square. The reconfiguration problem for Hamiltonian cycle or path in a sold
grid graph G asks the following question: given two Hamiltonian cycles (or paths)
of G, can we transform one cycle (or path) to the other using some "operation"
such that we get a Hamiltonian cycle (or path) of G in the intermediate steps (i.e.,
after each application of the operation)? In this thesis, we investigate reconfiguration
problems for Hamiltonian cycles and paths in the context of two types of solid
graphs: rectangular grid graphs, which have a rectangular outer boundary, and L-
shaped grid graphs, which have a single reflex corner on the outer boundary, under
three operations we define, flip, transpose and switch, that are local in the grid.
Reconfiguration of Hamiltonian cycles and paths in embedded grid graphs has
potential applications in path planning, robot navigation, minimizing turn costs in
milling problems, minimizing angle costs in TSP, additive manufacturing and 3D
printing, and in polymer science.
In this thesis, we introduce a complexity measure called bend complexity for Hamiltonian
paths and cycles in grid graphs, and using those measures we measure complexity
of a grid graph G and give upper and lower bounds on the maximum bend
complexity of an mxn grid graph. We define three local operations, flip, transpose and switch, where local means that the operations are applied on vertices that are close in
the grid graph but may not be close on the path or cycle. We show that any Hamiltonian
cycle or path can be reconfigured to any other Hamiltonian cycle or path in an mxn
rectangular grid graph, where m <= 4, using O(|G|) flips and transposes, regardless of
the bend complexities of the two cycles. We give algorithms to reconfigure 1-complex
Hamiltonian cycles in a rectangular or L-shaped grid graph G using O(|G|) flips and
transposes, where the intermediate steps are also 1-complex Hamiltonian cycles. Finally,
we establish the structure of 1-complex Hamiltonian paths between diagonally
opposite corners s and t of a rectangular grid graph, and then provide a strategy,
based on work in progress, for designing an algorithm to reconfigure between any two
1-complex s, t Hamiltonian paths using switch operations.
Description
Keywords
Hamiltonian cycle, Hamiltonian path, Grid graph, Reconfiguration, Local operation, Algorithm