A graph G is a k-dot product graph if there exists a vector labelling u : V (G)→ Rk such that u(i)Tu(j) ≥ 1 if and only if ij ∈ E(G). Fiduccia, Scheinerman, Trenk and Zito [4] asked whether every planar graph is a 3-dot product graph. We show that the answer is “no”. On the other hand, every planar graph is a 4-dot product graph. We also answer the corresponding questions for planar graphs of p...

Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every planar graph $G$ there is a $H$ with treewidth at most 8 path $P$ such $G\subseteq H\boxtimes P$. We improve this result by replacing "treewidth 8" "simple 6".

The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. An and Wu introduce the notion of list linear arboricity lla(G) of a graph G and conjecture that lla(G) = la(G) for any graph G. We confirm that this conjecture is true for any planar graph having ∆ > 13, or for any planar graph with ∆ > 7 and without i-cycles for some i ∈ {3, 4, 5}....

Algorithms for the construction of spanning planar subgraphs of Unit Disk Graphs (UDGs) do not ensure connectivity of the resulting graph under single edge deletion. To overcome this deficiency, in this paper we address the problem of augmenting the edge set of planar geometric graphs with straight line edges of bounded length so that the resulting graph is planar and 2-edge connected. We give ...

It is shown that every connected planar straight line graph with n ≥ 3 vertices has an embedding preserving augmentation to a 2-edge connected planar straight line graph with at most b(2n − 2)/3c new edges. It is also shown that every planar straight line tree with n ≥ 3 vertices has an embedding preserving augmentation to a 2-edge connected planar topological graph with at most bn/2c new edges...