Strong resolvability in product graphs

En aquesta tesi s'estudia la dimensio metrica forta de grafs producte. Els resultats mes importants de la tesi se centren en la recerca de relacions entre la dimensio metrica forta de grafs producte i la dels seus factors, juntament amb altres invariants d'aquests factors. Aixi, s'han estudiat els seguents productes de grafs: producte cartesia, producte directe, producte fort, producte lexicografic, producte corona, grafs unio, suma cartesiana, i producte arrel, d'ara endavant "grafs producte". Hem obtingut formules tancades per la dimensio metrica forta de diverses families no trivials de grafs producte que inclouen, per exemple, grafs bipartits, grafs vertexs transitius, grafs hamiltonians, arbres, cicles, grafs complets, etc, i hem donat fites inferiors i superiors generals, expressades en termes d'invariants dels grafs factors, com ara, l'ordre, el nombre d'independencia, el nombre de cobriment de vertexs, el nombre d'aparellament, la connectivitat algebraica, el nombre de clique, i el nombre de clique lliure de bessons. Tambe hem descrit algunes classes de grafs producte, on s'assoleixen aquestes fites. Es conegut que el problema de trobar la dimensio metrica forta d'un graf connex es pot transformar en el problema de trobar el nombre de cobriment de vertexs de la seva corresponent graf de resolubilitat forta. En aquesta tesi hem aprofitat aquesta eina i hem trobat diverses relacions entre el graf de resolubilitat forta de grafs producte i els grafs de resolubilitat forta dels seus factors. Per exemple, es notable destacar que el graf de resolubilitat forta del producte cartesia de dos grafs es isomorf al producte directe dels grafs de resolubilitat forta dels seus factors.

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