Some invariants related to threshold and chain graphs

Abstract

Let $ G = (V, E) $ be a finite simple connected graph. We say a graph $ G $ realizes a code of the type $ 0^{s_1}1^{t_1}0^{s_2}1^{t_2}\cdots0^{s_k}1^{t_k} $ if and only if $ G $ can be obtained from the code by some rule. Some classes of graphs such as threshold and chain graphs realizes a code of the type mentioned above. The main objective of this research article is to develop some computationally feasible methods to determine some interesting graph theoretical invariants. We present an efficient algorithm to determine the metric dimension of threshold and chain graphs. We compute threshold dimension and restricted threshold dimension of threshold graphs. We discuss $ L(2, 1) $-coloring in threshold and chain graphs. In fact, for every threshold graph $ G $, we establish a formula by which we can obtain the $ \lambda $-chromatic number of $ G $. Finally, we provide an algorithm to compute the $ \lambda $-chromatic number of chain graphs.