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- W1670812079 abstract "Let $G=(V,E)$ be a bipartite graph embedded in a plane (or $n$-holed torus). Two subgraphs of $G$ differ by a {it $Z$-transformation} if their symmetric difference consists of the boundary edges of a single face---and if each subgraph contains an alternating set of the edges of that face. For a given $phi: V mapsto mathbb Z^+$, $S_{phi}$ is the set of subgraphs of $G$ in which each $vin V$ has degree $phi(v)$. Two elements of $S_{phi}$ are said to be adjacent if they differ by a $Z$-transformation. We determine the connected components of $S_{phi}$ and assign a {it height function} to each of its elements. If $phi$ is identically two, and $G$ is a grid graph, $S_{phi}$ contains the partitions of the vertices of $G$ into cycles. We prove that we can always apply a series of $Z$-transformations to decrease the total number of cycles provided there is enough ``slack'' in the corresponding height function. This allows us to determine in polynomial time the minimal number of cycles into which $G$ can be partitioned provided $G$ has a limited number of non-square faces. In particular, we determine the Hamiltonicity of polyomino graphs in $O(|V|^2)$ steps. The algorithm extends to $n$-holed-torus-embedded graphs that have grid-like properties. We also provide Markov chains for sampling and approximately counting the Hamiltonian cycles of $G$." @default.
- W1670812079 created "2016-06-24" @default.
- W1670812079 creator A5074134351 @default.
- W1670812079 date "2000-08-31" @default.
- W1670812079 modified "2023-09-27" @default.
- W1670812079 title "Computing and Sampling Restricted Vertex Degree Subgraphs and Hamiltonian Cycles" @default.
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