On the algorithmic complexity of twelve covering and independence parameters of graphs

The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs

Centroidal bases in graphs

We introduce the notion of a centroidal locating set of a graph $G$, that is, a set $L$ of vertices such that all vertices in $G$ are uniquely determined by their relative distances to the vertices of $L$. A centroidal locating set of $G$ of minimum size is called a centroidal basis, and its size is the centroidal dimension $CD(G)$. This notion, which is related to previous concepts, gives a new way of identifying the vertices of a graph. The centroidal dimension of a graph $G$ is lower- and upper-bounded by the metric dimension and twice the location-domination number of $G$, respectively. The latter two parameters are standard and well-studied notions in the field of graph identification. We show that for any graph $G$ with $n$ vertices and maximum degree at least~2, $(1+o(1))\frac{\ln n}{\ln\ln n}\leq CD(G) \leq n-1$. We discuss the tightness of these bounds and in particular, we characterize the set of graphs reaching the upper bound. We then show that for graphs in which every pair of vertices is connected via a bounded number of paths, $CD(G)=\Omega\left(\sqrt{|E(G)|}\right)$, the bound being tight for paths and cycles. We finally investigate the computational complexity of determining $CD(G)$ for an input graph $G$, showing that the problem is hard and cannot even be approximated efficiently up to a factor of $o(\log n)$. We also give an $O\left(\sqrt{n\ln n}\right)$-approximation algorithm