4 research outputs found
Finding Paths in Graphs Avoiding Forbidden Transitions;
Let v be a vertex of a graph G; a transition graph T(v) of v is a graph whose vertices are the edges incident with v. We consider graphs G with prescribed transition systems T={T(v) | vV(G)}. A path P in G is called T-compatible, if each pair uv,vw of consecutive edges of P form an edge in T(v). Let be a given class of graphs (closed under isomorphism). We study the computational complexity of finding T-compatible paths between two given vertices of a graph for a specified transition system . Our main result is that a dichotomy holds (subject to the assumption Pâ NP). That is, for a considered class , the problem is either (1) NP-complete, or (2) it can be solved in linear time. We give a criterionâbased on vertex induced subgraphsâwhich decides whether (1) or (2) holds for any given class
Autour de la connexité dans les graphes avec conflits
We will look at graphs with conflicts (conflict is a pair of edges can not simultaneously be part of the same subgraph), in which we will study different types of problems related to the existence of subgraphs without conflict. The nature of the problems is both combinatorial and algorithmic. Our guideline is the notion of connectivity. We will see several results, simple without conflict, are no longer when adding conflicts. We will present exact algorithms (not polynomial), \mathcal{N P}-completeness results and sufficient conditions ensuring the existence of certain objects (spanning tree, path and Hamiltonian cycle) without conflict.Nous nous intĂ©resserons aux graphes avec conflits (un conflit est une paire dâarĂȘtes ne pouvant pas simultanĂ©ment faire partie dâun mĂȘme sous-graphe), dans lesquels nous Ă©tudierons diffĂ©rents types de problĂšmes liĂ©s Ă lâexistence de sous-graphes sans conflit, de nature aussi bien algorithmique que combinatoire, notre ligne directrice Ă©tant la notion de connectivitĂ©. Nous verrons que plusieurs rĂ©sultats, simples sans conflit, ne le sont plus lors de lâajout de conflits. Nous prĂ©senterons : des algorithmes exacts (non polynomiaux), des rĂ©sultats de \mathcal{N P}-complĂ©tude, et des conditions suffisantes assurant lâexistence de certains objets (arbre couvrant, chemin et cycle hamiltonien) sans conflits