# On the 2-independence subdivision number of graphs

Document Type : Original paper

Authors

1 Department of Mathematics University of Blida B.P. 270, Blida, Algeria

2 Department of Mathematics, University of Blida 1. B.P. 270, Blida, Algeria

3 Department of Mathematics University of Blida 1, B.P. 270, Blida, Algeria

Abstract

A subset $S$ of vertices in a graph $G=(V,E)$ is $2$-independent if every
vertex of $S$ has at most one neighbor in $S.$ The $2$-independence number
is the maximum cardinality of a $2$-independent set of $G.$ In this paper,
we initiate the study of the $2$-independence subdivision number $\mathrm{sd}% _{\beta _{2}}(G)$ defined as the minimum number of edges that must be
subdivided (each edge in $G$ can be subdivided at most once) in order to
increase the $2$-independence number. We first show that for every connected
graph $G$ of order at least three, $1\leq \mathrm{sd}_{\beta _{2}}(G)\leq 2,$
and we give a necessary and sufficient condition for graphs $G$ attaining
each bound. Moreover, restricted to the class of trees, we provide a
constructive characterization of all trees $T$ with $\mathrm{sd}_{\beta _{2}}(T)=2,$ and we show that such a characterization suggests an algorithm
that determines whether a tree $T$\ has\textrm{\ }$\mathrm{sd}_{\beta _{2}}(T)=2$\ or $\mathrm{sd}_{\beta _{2}}(T)=1$\ in polynomial time.

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Main Subjects

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