### Abstract

For digraphs G and H, a homomorphism of G to H is a mapping f : V(G)→V(H) such that uv ε A(G) implies f(u)f(v) ε A(H). If, moreover, each vertex u ε V(G) is associated with costs C_{i}(u), i ε V(H), then the cost of a homomorphism / is ∑_{u}εv(G) c_{f(u)}(u). For each fixed digraph H, the minimum cost homomorphism problem for H, denoted MinHOM(H), can be formulated as follows: Given an input digraph G, together with costs C_{i}(u), u ε V(G), i ε V(H), decide whether there exists a homomorphism of G to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well-studied optimization problems such as the minimum cost chromatic partition and repair analysis problems. We focus on the minimum cost homomorphism problem for locally semicomplete digraphs and quasi-transitive digraphs which are two well-known generalizations of tournaments. Using graph-theoretic characterization results for the two digraph classes, we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for both classes.

Original language | English |
---|---|

Pages (from-to) | 227-232 |

Number of pages | 6 |

Journal | Australasian Journal of Combinatorics |

Volume | 46 |

Publication status | Published - Feb 2010 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Australasian Journal of Combinatorics*,

*46*, 227-232.

**Minimum cost homomorphisms to locally semicomplete digraphs and quasi-transitive digraphs.** / Gupta, A.; Gutin, G.; Karimi, M.; Kim, E. J.; Rafiey, A.

Research output: Contribution to journal › Article

*Australasian Journal of Combinatorics*, vol. 46, pp. 227-232.

}

TY - JOUR

T1 - Minimum cost homomorphisms to locally semicomplete digraphs and quasi-transitive digraphs

AU - Gupta, A.

AU - Gutin, G.

AU - Karimi, M.

AU - Kim, E. J.

AU - Rafiey, A.

PY - 2010/2

Y1 - 2010/2

N2 - For digraphs G and H, a homomorphism of G to H is a mapping f : V(G)→V(H) such that uv ε A(G) implies f(u)f(v) ε A(H). If, moreover, each vertex u ε V(G) is associated with costs Ci(u), i ε V(H), then the cost of a homomorphism / is ∑uεv(G) cf(u)(u). For each fixed digraph H, the minimum cost homomorphism problem for H, denoted MinHOM(H), can be formulated as follows: Given an input digraph G, together with costs Ci(u), u ε V(G), i ε V(H), decide whether there exists a homomorphism of G to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well-studied optimization problems such as the minimum cost chromatic partition and repair analysis problems. We focus on the minimum cost homomorphism problem for locally semicomplete digraphs and quasi-transitive digraphs which are two well-known generalizations of tournaments. Using graph-theoretic characterization results for the two digraph classes, we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for both classes.

AB - For digraphs G and H, a homomorphism of G to H is a mapping f : V(G)→V(H) such that uv ε A(G) implies f(u)f(v) ε A(H). If, moreover, each vertex u ε V(G) is associated with costs Ci(u), i ε V(H), then the cost of a homomorphism / is ∑uεv(G) cf(u)(u). For each fixed digraph H, the minimum cost homomorphism problem for H, denoted MinHOM(H), can be formulated as follows: Given an input digraph G, together with costs Ci(u), u ε V(G), i ε V(H), decide whether there exists a homomorphism of G to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well-studied optimization problems such as the minimum cost chromatic partition and repair analysis problems. We focus on the minimum cost homomorphism problem for locally semicomplete digraphs and quasi-transitive digraphs which are two well-known generalizations of tournaments. Using graph-theoretic characterization results for the two digraph classes, we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for both classes.

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M3 - Article

VL - 46

SP - 227

EP - 232

JO - Australasian Journal of Combinatorics

JF - Australasian Journal of Combinatorics

SN - 1034-4942

ER -