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Getting new algorithmic results by extending distance-hereditary graphs via split composition.

Serafino Cicerone1, Gabriele Di Stefano1

  • 1Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, L'Aquila, Italy.

Peerj. Computer Science
|July 26, 2021
PubMed
Summary
This summary is machine-generated.

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This paper introduces a new graph class, Gen(∗;P3,C3,C5), extending distance-hereditary graphs. Efficient algorithms are developed for key combinatorial problems within this class.

Area of Science:

  • Graph Theory
  • Discrete Mathematics
  • Computer Science

Background:

  • The study builds upon the established class of distance-hereditary graphs, denoted as Gen(∗;P3,C3).
  • It introduces the concept of stretch number to establish relationships within graph hierarchies, specifically DH(k) where DH(1) represents distance-hereditary graphs.

Purpose of the Study:

  • To define and investigate a new graph class, Gen(∗;P3,C3,C5), generated by specific composition operations.
  • To establish efficient algorithms for fundamental combinatorial problems on this new graph class.
  • To explore the properties of this graph class, including its relationship to distance-hereditary graphs and its clique-width.

Main Methods:

  • Utilizing the split composition operation with path P3, cycle C3, and cycle C5 as components to define the graph class.
Keywords:
Distance-hereditary graphsGraph algorithmsGraph classesSplit decompositionStretch number

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  • Applying the concept of stretch number to analyze the relationship between Gen(∗;P3,C3) and the DH(k) hierarchy.
  • Developing and proving the existence of efficient algorithms for recognition, stretch number, stability number, clique number, domination number, chromatic number, and graph isomorphism.

    • Main Results:

    • The graph class Gen(∗;P3,C3,C5) is formally defined and characterized.
    • Efficient algorithms are established for solving several key combinatorial problems within this class.
    • It is proven that graphs within Gen(∗;P3,C3,C5) possess bounded clique-width.

    Conclusions:

    • The newly defined graph class Gen(∗;P3,C3,C5) offers a valuable extension to distance-hereditary graphs.
    • The development of efficient algorithms facilitates practical applications and further research in graph theory.
    • The bounded clique-width property suggests potential for efficient graph processing and representation.