## TOTAL VERTEX IRREGULARITY STRENGTH OF INTERVAL GRAPHS

## TOTAL VERTEX IRREGULARITY STRENGTH OF INTERVAL GRAPHS

*A. Rana *

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## Abstract

A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers (usually positive integers). For a simple graph G = (V,E) with vertex set V and edge set E, a labeling φ : V ∪ E → {1, 2, ..., k} is called total k-labeling. The associated vertex weight of a vertex x ∈ V (G) under a total k-labeling φ is defined as wt(x) = φ(x)+ E φ(xy) where N(x) is the set of neighbors of the vertex x. A total y∈N(x) k-labeling is defined to be a vertex irregular total labeling of a graph G, if wt(x) ̸= wt(y) holds for every two different vertices x and y of G. The minimum k for which a graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G). In this paper, total vertex irregularity strength of interval graphs is studied. In particular, an efficient algorithm is designed to compute tvs of proper interval graphs and bounds of tvs are presented for interval graphs.

## Keywords

Interval graphs, vertex irregular total labeling, total vertex irregularity strength, design of algorithms.