Is there simple reduction Dominating Set to Vertex Cover? Is there simple reduction Dominating Set to Vertex Cover?


In the other direction the reduction is simple.

Searching the web returned blog.

It warns This is not finished yet and experiments suggest
the reduction doesn't work.



Added by "simple" I mean graph transformation
$G \to G'$, s.t. $G$ has DS of size $k$ iff $G'$
has VC of size $f(G,k)$ and the transformation
does not depend on $k$ (to avoid reduction to SAT).
How about the following transformation: Given a graph $G(V,E)$ with $V=\{1,\ldots,n\}$ construct $G'(V',E')$ as follows: $V'$ initially consists of two sets of vertices: the vertices $u_i, 1\le i\le n$ and the vertices $v_{i,j}, 1\le i,j\le n$ (a total of $n^2+n$ vertices). We add the following sets of edges: first, connect all $v_{i,j}$ with $v_{i,k}$ for $k\neq j$, so the graph now consists of $n$ cliques of order $n$ and $n$ isolated vertices. Next, for all $i$, connect $u_i$ with the vertex $v_{i,i}$. Then, for all $i$ connect $u_i$ with all vertices $v_{j,i}$ for which $(i,j)\in E$. On the other hand, for all $i,j$ such that $(i,j)\not\in E$ construct a new vertex $v'_{i,j}$ and connect it to $v_{i,j}$ (this constructs two new degree-1 vertices for each non-edge of $G$). This completes the construction.

The claim is that $G'$ has a vertex cover of size $n(n-1)+k$ if and only if $G$ has a dominating set of size $k$. The intuition is that any vertex cover must take at least $n-1$ vertices from each clique (giving at least $n(n-1)$ vertices overall), allowing us to select exactly $k$ of the $u_i$ vertices to represent the dominating set. The vertex that is left out of the $i$-th clique encodes the vertex of the original graph that was used to dominate $i$. Observe that the leaves $v'_{i,j}$ ensure that vertices corresponding to non-edges are not left out of the vertex cover. 

Thus, the direction dominating set of $G$ $\to$ vertex cover of $G'$ is straightforward: select the vertices from $u_i$ that represent the dominating set and from each clique select all vertices except the one that is adjacent to a selected $u_i$. For the converse direction, first note that it is never optimal to take a $v'_{i,j}$ vertex in the vertex cover since these vertices have degree 1, so their neighbors must be selected. Second, observe that in any vertex cover that selects all $n$ vertices of a clique we can exchange one of them with its neighboring $u_i$ vertex, therefore we can assume that an optimal vertex cover selects $k$ such $u_i$ vertices. These must be a dominating set of $G$.