DeepAI AI Chat
Log In Sign Up

Find Subtrees of Specified Weight and Cycles of Specified Length in Linear Time

by   On-Hei Solomon Lo, et al.

We introduce a variant of DFS which finds subtrees of specified weight in linear time, by which, as observed by Mohr, cycles of specified length in planar hamiltonian graphs can be found. We show, for example, that every planar hamiltonian graph G with minimum degree δ≥ 4 has a cycle of length k for every k ∈{|V(G)|/2, ..., |V(G)|/2 + 3} with 3 ≤ k ≤ |V(G)|, that every planar hamiltonian graph G with δ≥ 5 has a cycle of length k for every k ∈{1/4|V(G)|, ..., 3/4|V(G)| + 3} with 3 ≤ k ≤ |V(G)|, and that if |V(G)| ≥ 8 is even, every 3-connected planar hamiltonian graph G with δ≥ 4 has a cycle of length |V(G)|/2 - 1 or |V(G)|/2 - 2. Each of these cycles can be found in linear time if a Hamilton cycle of the graph is given. Another interesting consequence follows from our tool is that, given an instance of the number partitioning problem, i.e. a multiset of positive integers {a_1, ..., a_N}, if ∑_i = 1^N a_i ≤ 2N - 2, then a partition always exists and can be found in linear time.


page 1

page 2

page 3

page 4


Are highly connected 1-planar graphs Hamiltonian?

It is well-known that every planar 4-connected graph has a Hamiltonian c...

Exact and Approximate Algorithms for Computing a Second Hamiltonian Cycle

In this paper we consider the following total functional problem: Given ...

Planar graphs without pairwise adjacent 3-,4-,5-, and 6-cycle are 4-choosable

Xu and Wu proved that if every 5-cycle of a planar graph G is not simult...

Model-Based Diagnosis using Structured System Descriptions

This paper presents a comprehensive approach for model-based diagnosis w...

Port-Hamiltonian Realizations of Linear Time Invariant Systems

The question when a general linear time invariant control system is equi...

Flip distances between graph orientations

Flip graphs are a ubiquitous class of graphs, which encode relations ind...

A Linearly-growing Conversion from the Set Splitting Problem to the Directed Hamiltonian Cycle Problem

We consider a direct conversion of the, classical, set splitting problem...