## Abstract

A graph G is claw-free if no induced subgraph of it is isomorphic to the complete bipartite graph K _{1,3}, and it is prime if |V (G)| ≥ 4 and there is no X V (G) with 1 < |X| < |V (G)| such that every vertex of V (G) \ X with a neighbor in X is adjacent to every vertex of X. In particular, if G is prime, then both G and G ^{c} are connected. This paper has two main results. The first one is that if G is a prime graph that is not a member of a particular family of exceptions, and H is a prime induced subgraph of G, then (up to isomorphism) G can be grown from H, adding one vertex at a time, in such a way that all the graphs constructed along the way are prime induced subgraphs of G. A simplicial clique in G is a nonempty clique K such that for every k ∈ K the set of neighbors of k in V (G) \ K is a clique. Our second result is that a prime claw-free graph G has at most |V (G)| + 1 simplicial cliques, and we give an algorithm to find them all with running time O(|V (G)| ^{4}). In particular, this answers a question of Prasad Tetali [private communication] who asked if there is an efficient algorithm to test if a claw-free graph has a simplicial clique. Finally, we apply our results to claw-free graphs that are not prime. Such a graph may have exponentially many simplicial cliques, so we cannot list them all in polynomial time, but we can in a sense describe them.

Original language | English (US) |
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Pages (from-to) | 860-880 |

Number of pages | 21 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 26 |

Issue number | 2 |

DOIs | |

State | Published - Sep 6 2012 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Keywords

- Claw-free graphs
- Induced subgraphs
- Prime graphs
- Simplicial clique
- Splitter theorem