Excluding an induced wheel minor in graphs without large induced stars

Abstract

We study a conjecture due to Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht stating that for any positive integer d and any planar graph H, the class of all K_{1,d}-free graphs without H as an induced minor has bounded tree-independence number. A k-wheel is the graph obtained from a cycle of length k by adding a vertex adjacent to all vertices of the cycle. We show that the conjecture of Dallard et al. is true when H is a k-wheel for any k at least 3. Our proof uses a generalization of the concept of brambles to tree-independence number. As a consequence of our main result, several important NP-hard problems such as Maximum Independent Set are tractable on K_{1,d}-free graphs without large induced wheel minors. Moreover, for fixed d and k, we provide a polynomial-time algorithm that, given a K_{1,d}-free graph G as input, finds an induced minor model of a k-wheel in G if one exists.V prispevku preučujemo domnevo Dallarda, Krnca, Kwona, Milaniča, Munara, Štorgla in Wiederrechta, ki trdi, da ima za poljubno pozitivno celo število d in poljuben ravninski graf H razred vseh grafov brez inducirane zvezde K_{1,d} in brez induciranega minorja izomorfnega grafu H omejeno drevesno neodvisnostno število. Za celo število k vsaj 3 je k-kolo graf, ki ga dobimo iz cikla dolžine k z dodajanjem točke, ki sosednja vsem točkam cikla. Pokažemo, da domneva Dallarda idr. velja za primer, ko je H k-kolo za poljubno celo število k vsaj 3. Naš dokaz uporablja posplošitev koncepta trnjevja (ang. bramble) na drevesno neodvisnostno število. Posledica našega glavnega rezultata je, da je več pomembnih NP-težkih problemov, kot je največja neodvisna množica, polinomsko rešljivih na grafih brez inducirane fiksne zvezde in brez induciranih minorjev, izomorfnih nekemu fiksnemu kolesu. Poleg tega za fiksna d in k razvijemo polinomski algoritem, ki ob danem vhodnem grafu G brez inducirane zvezde K_{1,d} poišče model induciranega minorja za k-kolo v grafu G, če ta obstaja