Classes of graphs with no long cycle as a vertex-minor are polynomially χ-bounded

Abstract

A class G of graphs is χ-bounded if there is a function f such that for every graph G P G and every induced subgraph H of G, χpHq ď f pωpHqq. In addition, we say that G is polynomially χ-bounded if f can be taken as a polynomial function. We prove that for every integer n ě 3, there exists a polynomial f such that χpGq ď f pωpGqq for all graphs with no vertex-minor isomorphic to the cycle graph Cn. To prove this, we show that if G is polynomially χ-bounded, then so is the closure of G under taking the 1-join operation.

Graphs of bounded cliquewidth are polynomially $χ$-bounded

“…It seems that our main result may be applicable for establishing polynomial χ-boundedness of classes defined by forbidding fixed vertex-minors, similarly to the work of Dvořák and Král' [DK12]; we discuss these connections in Section 5.2. In particular, we generalize two related statements that were used in this context: the result of Chudnovsky et al [CPST13] that the closure of a polynomially χ-bounded class under the substitution operation is also polynomially χ-bounded, and the recent result of Kim et al [KKOS19] that the same holds also for the operation of taking 1-joins. Indeed, these two cases follow from taking k = 1 and k = 2 in our main theorem, respectively.…”

“…Note here that by using the second part of statement of Corollary 5.4 and the recent result of Davies and McCarty that circle graphs are quadratically χ-bounded [DM19], we can in the same manner argue that W 5 -vertex-minor-free graphs are polynomially χ-bounded. In a similar manner -by using the second part of statement of Corollary 5.4 together with a suitable decomposition result using 1-joins -Kim et al [KKOS19] proved that the class of C -vertex-minor-free graphs is polynomially χ-bounded, for every 3.…”

“…Similarly, our Theorem 2.5 reduces proving polynomial χ-boundedness of D to proving polynomial χ-boundedness of C. Together with the quadratic χ-boundedness of circle graphs [DM19], this suggests that in Conjecture 5.5 one might expect even polynomial χ-boundedness. This question was also posed by Kim et al, see [KKOS19,Question 1.4]. Very recently, Geelen et al [GKMW19] proved that for any circle graph H, the class of graphs that exclude H as a vertex-minor has bounded rankwidth.…”

“…The first part of Corollary 5.4 was explicitely proved by Dvořák and Král' in [DK12], while the second was proved by Kim et al [KKOS19] using an argument tailored to 1-joins; hence, this is not a new result.…”

“…We would like to express our gratitude to Rose McCarty for communicating the problem to us and for bringing the works of Chudnovsky et al [CPST13] and of Kim et al [KKOS19] to our attention. We thank the anonymous reviewers for their efficiency and their help in improving the presentation of this paper.…”