An \emph{s-graph} is a graph with two kinds of edges: \emph{subdivisible} edges and \emph{real} edges. A \emph{realisation} of an s-graph $B$ is any graph obtained by subdividing subdivisible edges of $B$ into paths of arbitrary length (at least one). Given an s-graph $B$, we study the decision problem $\Pi_B$ whose instance is a graph $G$ and question is "Does $G$ contain a realisation of $B$ as an induced subgraph?". For several $B$'s, the complexity of $\Pi_B$ is known and here we give the complexity for several more. Our NP-completeness proofs for $\Pi_B$'s rely on the NP-completeness proof of the following problem. Let $\cal S$ be a set of graphs and $d$ be an integer. Let $\Gamma_{\cal S}^d$ be the problem whose instance is $(G, x, y)$ where $G$ is a graph whose maximum degree is at most d, with no induced subgraph in $\cal S$ and $x, y \in V(G)$ are two non-adjacent vertices of degree 2. The question is "Does $G$ contain an induced cycle passing through $x, y$?". Among several results, we prove that $\Gamma^3_{\emptyset}$ is NP-complete. We give a simple criterion on a connected graph $H$ to decide whether $\Gamma^{+\infty}_{\{H\}}$ is polynomial or NP-complete. The polynomial cases rely on the algorithm three-in-a-tree, due to Chudnovsky and Seymour.Comment: arXiv admin note: text overlap with arXiv:1308.667

A contact representation by triangles of a graph is a set of triangles in the plane such that two triangles intersect on at most one point, each triangle represents a vertex of the graph and two triangles intersects if and only if their corresponding vertices are adjacent. De Fraysseix, Ossona de Mendez and Rosenstiehl proved that every planar graph admits a contact representation by triangles. We strengthen this in terms of a simultaneous contact representation by triangles of a planar map and of its dual.A primal-dual contact representation by triangles of a planar map is a contact representation by triangles of the primal and a contact representation by triangles of the dual such that for every edge uv, bordering faces f and g, the intersection between the triangles corresponding to u and v is the same point as the intersection between the triangles corresponding to f and g. We prove that every 3-connected planar map admits a primal-dual contact representation by triangles. Moreover, the interiors of the triangles form a tiling of the triangle corresponding to the outer face and each contact point is a corner of exactly three triangles. Then we show that these representations are in one-to-one correspondence with generalized Schnyder woods defined by Felsner for 3-connected planar maps.