it is planar but adding any edge (on the given vertex set) would destroy that property. All faces (including the outer one) are then bounded by three edges, explaining the alternative term plane triangulation. The alternative names "triangular graph" or "triangulated graph" have also been used, but are ambiguous, as they more commonly refer to the line graph of a complete graph and to the chordal graphs respectively. Every maximal planar graph is a least 3-connected. If a maximal planar graph has vertices with , then it has precisely edges and faces. Apollonian networks are the maximal planar graphs formed by repeatedly splitting triangular faces into triples of smaller triangles. Equivalently, they are the planar 3-trees. Strangulated graphs are the graphs in which every peripheral cycle is a triangle. In a maximal planar graph (or more generally a polyhedral graph) the peripheral cycles are the faces, so maximal planar graphs are strangulated. The strangulated graphs include also the chordal graphs, and are exactly the graphs that can be formed by clique-sums (without deleting edges) of complete graphs and maximal planar graphs. Outerplanar graphs Outerplanar graphs are graphs with an embedding in the plane such that all vertices belong to the unbounded face of the embedding. Every outerplanar graph is planar, but the converse is not true: is planar but not outerplanar. A theorem similar to Kuratowski's states that a finite graph is outerplanar if and only if it does not contain a subdivision of or of . The above is a direct corollary of the fact that a graph is outerplanar if the graph formed from by adding a new vertex, with edges connecting it to all the other vertices, is a planar graph. A 1-outerplanar embedding of a graph is the same as an outerplanar embedding. For a planar embedding is -outerplanar if removing the vertices on the outer face results in a -outerplanar embedding. A graph is -outerplanar if it has a -outerplanar embedding. Halin graphs A Halin graph is a graph formed from an undirected plane tree (with no degree-two nodes) by connecting its leaves into a cycle, in the order given by the plane embedding of the tree. Equivalently, it is a polyhedral graph in which one face is adjacent to all the others. Every Halin graph is planar. Like outerplanar graphs, Halin graphs have low treewidth, making many algorithmic problems on them more easily solved