**Square-contact representations of partial 2-trees and triconnected simply-nested graphs**.

G. Da Lozza, W. E. Devanny, D. Eppstein, and T. Johnson.

arXiv:1710.00426.

*Proc. 28th Int. Symp. Algorithms and Computation (ISAAC 2017)*, Phuket, Thailand, 2017.

Leibniz International Proceedings in Informatics (LIPIcs) 92, pp. 24:1–24:16.

We show that the

*K*_{1,1,3}-free partial 2-trees and the Halin graphs other than*K*_{4}can all be represented as proper contact graphs of squares in the plane. Among partial 2-trees and Halin graphs, these are exactly the ones that can be embedded without nonempty triangles, which form an obstacle to the existence of square contact representations. However the graph of a square antiprism has no such representation despite being embeddable without any nonempty triangles.**Subexponential-time and FPT algorithms for embedded flat clustered planarity**.

G. Da Lozza, D. Eppstein, M. T. Goodrich, and S. Gupta.

arXiv:1803.05465

*Proc. 44th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2018)*, Lübbenau, Germany, to appear.

Clustered planarity is the problem of simultaneously drawing a planar graph and a clustering of its vertices (as Jordan curves surrounding the cluster) with no unnecessary crossings between edges or cluster boundaries; it remains unknown whether it can be solved in polynomial time. We provide parameterized and subexponential exact algorithms for the case where the planar embedding is fixed and the clusters form a partition of the vertices.

Co-authors – Publications – David Eppstein – Theory Group – Inf. & Comp. Sci. – UC Irvine

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