The Pendant Number of Line Graphs and Total Graphs

The parameter, pendant number of a graph G, is defined as the least number of end vertices of paths in a path decomposition of the given graph and is denoted as $$ \Pi _p(G) $$ . This paper determines the pendant number of regular graphs, complete r-partite graphs, line graphs, total graphs and line graphs of total graphs. We explore the bougainvillea graphs, e-pendant graphs and v-pendant graphs. If the pendant number is 2, then the number of paths in the path decomposition of the given graph is at most $$\Delta (G)$$ , the maximum degree of the graph. Hence, a large class of graphs give a more reasonable solution to Gallai's conjecture on number of paths in the given path decomposition.