Universal Geometric Graphs

We introduce and study the problem of constructing geometric graphs that have few vertices and edges and that are universal for planar graphs or for some sub-class of planar graphs; a geometric graph is \emph{universal} for a class $\mathcal H$ of planar graphs if it contains an embedding, i.e., a crossing-free drawing, of every graph in $\mathcal H$. Our main result is that there exists a geometric graph with $n$ vertices and $O(n \log n)$ edges that is universal for $n$-vertex forests; this extends to the geometric setting a well-known graph-theoretic result by Chung and Graham, which states that there exists an $n$-vertex graph with $O(n \log n)$ edges that contains every $n$-vertex forest as a subgraph. Our $O(n \log n)$ bound on the number of edges cannot be improved, even if more than $n$ vertices are allowed. We also prove that, for every positive integer $h$, every $n$-vertex convex geometric graph that is universal for $n$-vertex outerplanar graphs has a near-quadratic number of edges, namely $\Omega_h(n^{2-1/h})$; this almost matches the trivial $O(n^2)$ upper bound given by the $n$-vertex complete convex geometric graph. Finally, we prove that there exists an $n$-vertex convex geometric graph with $n$ vertices and $O(n \log n)$ edges that is universal for $n$-vertex caterpillars.Comment: 20 pages, 8 figures; a 12-page extended abstracts of this paper will appear in the Proceedings of the 46th Workshop on Graph-Theoretic Concepts in Computer Science (WG 2020

Noncanonical GLI1 signaling promotes stemness features and in vivo growth in lung adenocarcinoma

Aberrant Hedgehog/GLI signaling has been implicated in a diverse spectrum of human cancers, but its role in lung adenocarcinoma (LAC) is still under debate. We show that the downstream effector of the Hedgehog pathway, GLI1, is expressed in 76% of LACs, but in roughly half of these tumors, the canonical pathway activator, Smoothened, is expressed at low levels, possibly owing to epigenetic silencing. In LAC cells including the cancer stem cell compartment, we show that GLI1 is activated noncanonically by MAPK/ERK signaling. Different mechanisms can trigger the MAPK/ERK/GLI1 cascade including KRAS mutation and stimulation of NRP2 by VEGF produced by the cancer cells themselves in an autocrine loop or by stromal cells as paracrine cross talk. Suppression of GLI1, by silencing or drug-mediated, inhibits LAC cells proliferation, attenuates their stemness and increases their susceptibility to apoptosis in vitro and in vivo. These findings provide insight into the growth of LACs and point to GLI1 as a downstream effector for oncogenic pathways. Thus, strategies involving direct inhibition of GLI1 may be useful in the treatment of LACs