Abstract
For an arbitrary invariant ρ(G) of a graph G the ρ-vertex stability number vsρ(G) (ρ-edge stability number esρ(G)) is the minimum number of vertices (edges) whose removal results in a graph H⊆G with ρ(H)≠ρ(G). If such a vertex set (edge set) does not exist, then we set vsρ(G)=∞ (esρ(G)=∞).In the first part of this paper we give some general results for the ρ-vertex stability number. We prove among others a result which implies the well-known Theorem of Gallai on independence and vertex covering of graphs. In the second part we focus on the invariant chromatic number χ(G) and determine vsχ(G) for several classes of graphs. Moreover, we consider the relationship between esχ(G) and vsχ(G) and prove that their difference and their ratio may get arbitrarily large.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
