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In this paper, we investigate the approximability of two node deletionproblems. Given a vertex weighted graph $G=(V,E)$ and a specified, or"distinguished" vertex $p \in V$, MDD(min) is the problem of finding a minimumweight vertex set $S \subseteq V\setminus \{p\}$ such that $p$ becomes theminimum degree vertex in $G[V \setminus S]$; and MDD(max) is the problem offinding a minimum weight vertex set $S \subseteq V\setminus \{p\}$ such that$p$ becomes the maximum degree vertex in $G[V \setminus S]$. These are known$NP$-complete problems and have been studied from the parameterized complexitypoint of view in previous work. Here, we prove that for any $\epsilon > 0$,both the problems cannot be approximated within a factor $(1 - \epsilon)\logn$, unless $NP \subseteq DTIME(n^{\log\log n})$. We also show that for any$\epsilon > 0$, MDD(min) cannot be approximated within a factor $(1-\epsilon)\log n$ on bipartite graphs, unless $NP \subseteq DTIME(n^{\log\logn})$, and that for any $\epsilon > 0$, MDD(max) cannot be approximated within afactor $(1/2 - \epsilon)\log n$ on bipartite graphs, unless $NP \subseteqDTIME(n^{\log\log n})$. We give an $O(\log n)$ factor approximation algorithmfor MDD(max) on general graphs, provided the degree of $p$ is $O(\log n)$. Wethen show that if the degree of $p$ is $n-O(\log n)$, a similar result holdsfor MDD(min). We prove that MDD(max) is $APX$-complete on 3-regular unweightedgraphs and provide an approximation algorithm with ratio $1.583$ when $G$ is a3-regular unweighted graph. In addition, we show that MDD(min) can be solved inpolynomial time when $G$ is a regular graph of constant degree.

On the Complexity of Making a Distinguished Vertex Minimum or Maximum Degree by Vertex Deletion