Geodesic
The Saturation Number for the Length of Degree Monotone Paths
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 3, pp. 557-569.

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A degree monotone path in a graph G is a path P such that the sequence of degrees of the vertices in the order in which they appear on P is monotonic. The length (number of vertices) of the longest degree monotone path in G is denoted by mp(G). This parameter, inspired by the well-known Erdős- Szekeres theorem, has been studied by the authors in two earlier papers. Here we consider a saturation problem for the parameter mp(G). We call G saturated if, for every edge e added to G, mp(G + e) gt; mp(G), and we define h(n, k) to be the least possible number of edges in a saturated graph G on n vertices with mp(G) lt; k, while mp(G+e) ≥ k for every new edge e. We obtain linear lower and upper bounds for h(n, k), we determine exactly the values of h(n, k) for k = 3 and 4, and we present constructions of saturated graphs.
Keywords: paths, degrees, saturation 
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Caro, Yair; Lauri, Josef; Zarb, Christina. The Saturation Number for the Length of Degree Monotone Paths. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 3, pp. 557-569. http://geodesic.mathdoc.fr/item/DMGT_2015_35_3_a12/

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