Finding a kernel for d-Bounded degree deletion

Question

In $d$ Bounded degree deletion problem, we are given an undirected graph $G$ and a positive integer $k$, and the task is to find at most $k$ such vertices whose removal decreases the the maximum vertex degree of the graph to at most $d$.

The question is to how to find a polynomial kernel (in $k$ and $d$) for this problem.

I seem to be able to get the only reduction rule that if any vertex has degree $ > k+d$, it has to be there in the deletion set (if the answer to instance is yes). Because if it isn't, then at least $k+1$ of its neighbors have to be in deletion set. I can't seem to move beyond this point.

The exercise is from this book (exercise $2.9$).

I am also aware that we can remove edges between vertices with degree $< d$, and find solution in the modified graph (hint from the book). But I am not sure how it will be useful, in getting a bound over number of vertices/edges in $k$ and $d$.

I would appreciate only hints if possible (something maybe beyond the book hints).

PS: for $d=0$ this reduces to vertex cover problem.

Answer 1

Reduction Rule 1. Let $V$ be the set of vertices which are isolated. Convert the instance from $I = (G,k,d)$ to $I^{'} = (G -V, k,d)$. If $I^{'}$ is a yes instance, then so is $I$, because adding back the isolated vertices do not add on to the degree of other vertices. And isolated vertices have already have degree 0 ($\le d$ as $d \ge 0$). And if $I$ is a yes instance so is $I^{'}$, as $G-V$ is a sub-graph of $G$, and $k$ and $d$ remain same across the two instances.

Reduction Rule 2. If a vertex $v$ has degree $\ge k+d+1$ then it has to be kept in the deletion set. Otherwise $k+1$ of its neighbours have to be kept in the deletion set. Which cannot be done, as maximum size of deletion set is $k$. Thus after this reduction all vertices will have degree $\le k+d$.

Reduction Rule 3 If there are two vertices $v$ and $w$, such that degree of both of them is $\le d$. Then the edge $vw$ can be removed converting the instance from $I = (G, k,d)$ to $I^{'} = (G-\{vw\}, k,d)$. If $I$ is a yes instance so is $I^{'}$ as $G-\{vw\}$ is a sub-graph of $G$, with $k$ and $d$ remaining same across the instances. And if $I^{'}$ is a yes instance so is $I$, as adding back $vw$ can make the degree of $v$ and $w$ at most $d$ (as they initially had degree $\le d$).

Rule $1$ might be applied after applying Rule $3$ as well.

Now considering a graph where vertices with degree $\le d$ don't share an edge, no vertex is isolated and and all vertices have degree $\le k+d$ (ie. none of the above rules applied). Let $A$ be the set of vertices with degree $\le d$ and let $B$ be the set of vertices with degree $> d$. We can argue that $|B| \le k(k+d) +k$, otherwise the output is no. Because if $|B| > k(k+d) +k$, then as most $k$ elements will be chose from $B$ for the deletion set, and $ > k(k+d)$ elements will be left (in $B$). And as each of them has degree $> d$, they each have at least one edge going into the deletion set. But the deletion set can only accommodate $k(k+d)$ edges. Also as each vertex in $A$ has at least one neighbor (and all of them in $B$), the maximum number of vertices in $A$, will be bounded as $|A| \le |B|(k+d) \le (k(k+d)+k)(k+d) = k(k+d)(k+d+1)$. So $|A| + |B| \le k(k+d) + k(k+d)(k+d+1) = k(k+d)(k+d+2)$.