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 On pathos block line cut-vertex graph of a tree Commun. Korean Math. Soc. 2020 Vol. 35, No. 1, 1-12 https://doi.org/10.4134/CKMS.c170432Published online January 31, 2020 Hadonahalli Mudalagiraiah Nagesh PES University - Electronic City Campus Abstract : 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 Downloads: Full-text PDF   Full-text HTML