Domination Defect in Graphs: Guarding With Fewer Guards

• Published in: Indian journal of pure and applied mathematics Vol. 49; no. 2; pp. 349 - 364
• Main Authors: Das, Angsuman, Desormeaux, Wyatt J.
• Format: Journal Article
• Language: English
• Published: New Delhi Indian National Science Academy 01-06-2018; Springer Nature B.V
• Subjects: Applications of Mathematics; Defects; Graph theory; Graphs; Guards; Homomorphisms; Mathematics; Mathematics and Statistics; Numerical Analysis; vertex removal; domination defect; changing domination; Minimum dominating set; efficient dominating set
• ISSN: 0019-5588; 0975-7465
• DOI: 10.1007/s13226-018-0273-8

In this paper, we introduce a new graph parameter called the domination defect of a graph. The domination number γ of a graph
G
is the minimum number of vertices required to dominate the vertices of
G
. Due to the minimality of γ, if a set of vertices of
G
has cardinality less than γ then there are vertices of G that are not dominated by that set. The
k
-domination defect of
G
is the minimum number of vertices which are left un-dominated by a subset of γ -
k
vertices of
G
. We study different bounds on the
k
-domination defect of a graph
G
with respect to the domination number, order, degree sequence, graph homomorphisms and the existence of efficient dominating sets. We also characterize the graphs whose domination defect is 1 and find exact values of the domination defect for some particular classes of graphs.