On the Structure of Hamiltonian Graphs with Small Independence Number
arxiv(2024)
摘要
A Hamiltonian path (cycle) in a graph is a path (cycle, respectively) which
passes through all of its vertices. The problems of deciding the existence of a
Hamiltonian cycle (path) in an input graph are well known to be NP-complete,
and restricted classes of graphs which allow for their polynomial-time
solutions are intensively investigated. Until very recently the complexity was
open even for graphs of independence number at most 3. So far unpublished
result of Jedličková and Kratochvíl [arXiv:2309.09228] shows that
for every integer k, Hamiltonian path and cycle are polynomial-time solvable
in graphs of independence number bounded by k. As a companion structural
result, we determine explicit obstacles for the existence of a Hamiltonian path
for small values of k, namely for graphs of independence number 2, 3, and 4.
Identifying these obstacles in an input graph yields alternative
polynomial-time algorithms for Hamiltonian path and cycle with no large hidden
multiplicative constants.
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