## Abstract

An (h,s,t)-representation of a graph G consists of a collection of subtrees {S_{v}:v∈V(G)} of a tree T, such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, and (iii) there is an edge between two vertices in the graph if and only if the corresponding subtrees in T have at least t vertices in common. Jamison and Mulder denote the family of graphs that admit such a representation as [h,s,t]. Our main theorem shows that the class of weakly chordal graphs is incomparable with the class [h,s,t]. We introduce new characterizations of the graph K_{2,n} in terms of the families [h,s,2] and [h,s,3]. We then present our second main result characterizing the graphs in [4, 3, 2] as being the graphs in [4, 4, 2] avoiding a particular family of substructures, and we give a recognition algorithm for the family [4, 3, 2].

Original language | English |
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Pages (from-to) | 209-222 |

Number of pages | 14 |

Journal | Discrete Mathematics |

Volume | 340 |

Issue number | 2 |

DOIs | |

State | Published - 6 Feb 2017 |

### Bibliographical note

Funding Information:Support for the first author during the final stages of this research was provided in part by ISF grant 347/09 .

Publisher Copyright:

© 2016 Elsevier B.V.

## Keywords

- Chordal graph
- Complete bipartite graph
- Intersection graph of subtrees of a tree
- Weakly chordal graph

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics