Good characterizations and linear time recognition for 2-probe block graphs

Block graphs are graphs in which every block (biconnected component) is a clique. A graph $G=(V,E)$ is said to be an (unpartitioned) $k$-probe block graph if there exist $k$ independent sets $N_i\subseteq V$, $1\le i\le k$, such that the graph $G'$ obtained from $G$ by adding certain edges between vertices inside the sets $N_i$, $1\le i\le k$, is a block graph; if the independent sets $N_i$ are given, $G$ is called a partitioned $k$-probe block graph. In this paper we give good characterizations for $2$-probe block graphs, in both unpartitioned and partitioned cases. As an algorithmic implication, partitioned and unpartitioned probe block graphs can be recognized in linear time, improving a recognition algorithm of cubic time complexity previously obtained by Chang et al. [Block-graph width, Theoretical Computer Science 412 (2011), 2496--2502].