A Linear-Time Algorithm for Finding a Complete Graph Minor in a Dense Graph

Abstract

Let g(t) be the minimum number such that every graph G with average degree d(G) ≥ g(t) contains a K t -minor. Such a function is known to exist, as originally shown by Mader. Kostochka and Thomason independently proved that g(t) ∈ Θ(t √ log t). This article shows that for all fixed ǫ > 0 and fixed sufficiently large t ≥ t(ǫ), if d(G) ≥ (2 + ǫ)g(t) then we can find this K t -minor in linear time. This improves a previous result by Reed and Wood who gave a linear-time algorithm when d(G) ≥ 2 t−2 .

Minors in graphs of large θr-girth

“…This result has been proven by Kostochka in [20] and Thomason in [33] and a precise estimation of the constant c has been given by Thomason in [34]. For recent results related to conditions that force a clique minor see [13,15,19,22,23].…”

“…This result has been proven by Kostochka in [20] and Thomason in [33] and a precise estimation of the constant c has been given by Thomason in [34]. For recent results related to conditions that force a clique minor see [13,15,19,22,23].The girth of a graph G is the minimum length of a cycle in G. Interestingly, it follows that graphs of large minimum degree contain clique-minors whose order is an exponential function of their girth. In particular, it follows by the main result of Kühn and Osthus in [21] that there is a constant c such that, if a graph has minimum degree d ≥ 3 and girth z, then it contains as a minor a clique of size k, whereIn this paper we provide conditions, alternative to the above one, that can force the existence of a clique-minor whose size is exponential.H-girth.…”

Παραμετρικοί - προσεγγιστικοί αλγόριθμοι και ιδιότητες Erdős-Pósa σε γραφήματα

“…Το αποτέλεσμα αυτό απεδείχθη από τον Kostochka [41] και τον Thomason [60] και μία ακριβής εκτίμηση της σταθεράς c έχει δοθεί από τον Thomason [61]. Πιο πρόσφατα αποτελέσματα που αφορούν συνθήκες που εξαναγκάζουν την ύπαρξη μίας κλίκας ως έλασσον παρουσιάζονται στα [23,30,35,46,47].…”