par Fiorini, Samuel 
;Huynh, Tony ;Joret, Gwenaël ;Varvitsiotis, Antonios
Référence SIAM journal on discrete mathematics, 31, 1, page (438-453)
Publication Publié, 2017
;Huynh, Tony ;Joret, Gwenaël ;Varvitsiotis, AntoniosRéférence SIAM journal on discrete mathematics, 31, 1, page (438-453)
Publication Publié, 2017
Article révisé par les pairs
| Résumé : | Let G be a graph and p ϵ [1, oo]. The parameter fp(G) is the least integer k such that for all m and all vectors (rv)vϵV(G) ⊆ Rm, there exist vectors (qv)vϵV(G) ⊆ Rk satisfying ∥v - rw∥p = ∥qv - qw∥p for all vw ϵ E(G). It is easy to check that fp(G) is always finite and that it is minor monotone. By the graph minor theorem of Robertson and Seymour [J. Combin. Theory Ser. B, 92(2004), pp. 325-357], there are a finite number of excluded minors for the property fp(G) ≤ k. In this paper, we determine the complete set of excluded minors for f∞(G) ≤ 2. The two excluded minors are the wheel on five vertices and the graph obtained by gluing two copies of K4 along an edge and then deleting that edge. We also show that the same two graphs are the complete set of excluded minors for f1(G) ≤ 2. In addition, we give a family of examples that show that f∞, is unbounded on the class of planar graphs and f∞, is not bounded as a function of tree-width. |
