Geodesic
End Simplicial Vertices in Path Graphs
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 2, pp. 393-408

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A graph is a path graph if there is a tree, called UV -model, whose vertices are the maximal cliques of the graph and for each vertex x of the graph the set of maximal cliques that contains it induces a path in the tree. A graph is an interval graph if there is a UV -model that is a path, called an interval model. Gimbel [3] characterized those vertices in interval graphs for which there is some interval model where the interval corresponding to those vertices is an end interval. In this work, we give a characterization of those simplicial vertices x in path graphs for which there is some UV -model where the maximal clique containing x is a leaf in this UV -model.
Keywords: chordal graphs, clique trees, path graphs 
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Gutierrez, Marisa; Tondato, Silvia B. End Simplicial Vertices in Path Graphs. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 2, pp. 393-408. http://geodesic.mathdoc.fr/item/DMGT_2016_36_2_a10/