Outerplanar graph

In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. Outerplanar graphs may be characterized (analogously to Wagner's theorem for planar graphs) by the two forbidden minors K4 and K2,3, or by their Colin de Verdière graph invariants.They have Hamiltonian cycles if and only if they are biconnected, in which case the outer face forms the unique Hamiltonian cycle. Every outerplanar graph is 3-colorable, and has degeneracy and treewidth at most 2. The outerplanar graphs are a subset of the planar graphs, the subgraphs of series-parallel graphs, and the circle graphs. The maximal outerplanar graphs, those to which no more edges can be added while preserving outerplanarity, are also chordal graphs and visibility graphs. Outerplanar graphs were first studied and named by Chartrand & Harary (1967), in connection with the problem of determining the planarity of graphs formed by using a perfect matching to connect two copies of a base graph (for instance, many of the generalized Petersen graphs are formed in this way from two copies of a cycle graph). As they showed, when the base graph is biconnected, a graph constructed in this way is planar if and only if its base graph is outerplanar and the matching forms a dihedral permutation of its outer cycle. Chartrand and Harary also proved an analogue of Kuratowski's theorem for outerplanar graphs, that a graph is outerplanar if and only if it does not contain a subdivision of one of the two graphs K4 or K2,3. An outerplanar graph is an undirected graph that can be drawn in the plane without crossings in such a way that all of the vertices belong to the unbounded face of the drawing. That is, no vertex is totally surrounded by edges. Alternatively, a graph G is outerplanar if the graph formed from G by adding a new vertex, with edges connecting it to all the other vertices, is a planar graph. A maximal outerplanar graph is an outerplanar graph that cannot have any additional edges added to it while preserving outerplanarity. Every maximal outerplanar graph with n vertices has exactly 2n − 3 edges, and every bounded face of a maximal outerplanar graph is a triangle. Outerplanar graphs have a forbidden graph characterization analogous to Kuratowski's theorem and Wagner's theorem for planar graphs: a graph is outerplanar if and only if it does not contain a subdivision of the complete graph K4 or the complete bipartite graph K2,3. Alternatively, a graph is outerplanar if and only if it does not contain K4 or K2,3 as a minor, a graph obtained from it by deleting and contracting edges. A triangle-free graph is outerplanar if and only if it does not contain a subdivision of K2,3. A graph is outerplanar if and only if its Colin de Verdière graph invariant is at most two. The graphs characterized in a similar way by having Colin de Verdière invariant at most one, three, or four are respectively the linear forests, planar graphs, andlinklessly embeddable graphs.