Communications of the
Korean Mathematical Society CKMS

A \emph{pathos block line cut-vertex graph} of a tree $T$, written $PBL_{c}(T)$, is a graph whose vertices are the blocks, cut-vertices, and paths of a pathos of $T$, with two vertices of $PBL_{c}(T)$ adjacent whenever the corresponding blocks of $T$ have a vertex in common or the edge lies on the corresponding path of the pathos or one corresponds to a block $B_i$ of $T$ and the other corresponds to a cut-vertex $c_j$ of $T$ such that $c_j$ is in $B_i$; two distinct pathos vertices $P_m$ and $P_n$ of $PBL_{c}(T)$ are adjacent whenever the corresponding paths of the pathos $P_m(v_i, v_j)$ and $P_n(v_k, v_l)$ have a common vertex. We study the properties of $PBL_{c}(T)$ and present the characterization of graphs whose $PBL_{c}(T)$ are planar; outerplanar; maximal outerplanar; minimally nonouterplanar; eulerian; and hamiltonian. We further show that for any tree $T$, the crossing number of $PBL_{c}(T)$ can never be one.

Keywords: Crossing number, inner vertex number, path, cycle

MSC numbers: Primary 05C05, 05C45