Certifying Fully Dynamic Algorithms for Recognition and Hamiltonicity of Threshold and Chain Graphs

Solving problems on graphs dynamically calls for algorithms to function under repeated modifications to the graph and to be more efficient than solving the problem for the whole graph from scratch after each modification. Dynamic algorithms have been considered for several graph properties, for exam...

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Bibliographic Details
Published in:Algorithmica Vol. 85; no. 8; pp. 2454 - 2481
Main Authors: Beisegel, Jesse, Köhler, Ekkehard, Scheffler, Robert, Strehler, Martin
Format: Journal Article
Language:English
Published: New York Springer US 01.08.2023
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Apexes
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Graph theory
Graphs
Mathematics of Computing
Problem solving
Recognition
Shortest-path problems
Theory of Computation
05C45
Hamiltonian cycles
05C85
Chain graphs
Difference graphs
Hamiltonian paths
Fully dynamic algorithms
Threshold graphs
68R10
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:Solving problems on graphs dynamically calls for algorithms to function under repeated modifications to the graph and to be more efficient than solving the problem for the whole graph from scratch after each modification. Dynamic algorithms have been considered for several graph properties, for example connectivity, shortest paths and graph recognition. In this paper we present fully dynamic algorithms for the recognition of threshold graphs and chain graphs, which are optimal in the sense that the costs per modification are linear in the number of modified edges. Furthermore, our algorithms also consider the addition and deletion of sets of vertices as well as edges. In the negative case, i.e., where the graph is not a threshold graph or chain graph anymore, our algorithms return a certificate of constant size. Additionally, we present optimal fully dynamic algorithms for the Hamiltonian cycle problem and the Hamiltonian path problem on threshold and chain graphs which return a vertex cutset as certificate for the non-existence of such a path or cycle in the negative case.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-023-01107-1