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On the Structure of Hamiltonian Graphs with Small Independence Number

Nikola Jedličková, Jan Kratochvíl

Abstract

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 So far unpublished result of Jedličková and Kratochvíl [arXiv:2309.09228] shows that for every integer $k$, Hamiltonian path and cycle are polynomial-time solvable in graphs of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.

On the Structure of Hamiltonian Graphs with Small Independence Number

Abstract

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 So far unpublished result of Jedličková and Kratochvíl [arXiv:2309.09228] shows that for every integer , Hamiltonian path and cycle are polynomial-time solvable in graphs of independence number bounded by . As a companion structural result, we determine explicit obstacles for the existence of a Hamiltonian path for small values of , 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.
Paper Structure (7 sections, 9 theorems, 5 figures)

This paper contains 7 sections, 9 theorems, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hamiltonian paths in graphs with small independence number
  4. $3K_1$-free graphs
  5. $4K_1$-free graphs
  6. $5K_1$-free graphs
  7. Conclusion

Key Result

proposition thmcounterproposition

denley2001generalization Let $G$ be an $s$-connected graph with $s \geq 1$. If $x \in V(G)$, $Y \subseteq V(G)$ and $x \not \in Y$, then there exist distinct vertices $y_1, \ldots, y_m \in Y$, where $m= \min \{s, |Y|\}$, and internally disjoint paths $P_1, \ldots, P_m$ such that for every $i \in 1,\

Figures (5)

  • Figure 1: An illustration of violation of the conditions in Theorem \ref{['thm:4k1']}.
  • Figure 2: An illustration of violation of conditions (c),(d),(e) and (f) in Theorem \ref{['thm:4k1_1con']}. Dashed edge can be present or not.
  • Figure 3: An illustration of possible situations in $Q_2$ in Case 2c of Theorem \ref{['thm:4k1_1con']}.
  • Figure 4: An illustration of violations of conditions (a),(b),(c) and (d) in Theorem \ref{['thm:4k1pc']}. Dashed edges can be present or not.
  • Figure 5: An illustration of violations of the conditions (a),(b),(c) and (d) in Theorem \ref{['thm:5k1']}.

Theorems & Definitions (16)

  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • ...and 6 more