Separating an r-outerplanar graph into gluable pieces
نویسندگان
چکیده
LetG be an r-outerplanar graph with n vertices. We provide a sequence of log(n)/(r+ 1) +8r separators in G, each containing a fixed number (at most 2r) of integerlabeled vertices and each separating the graph in a well-defined left and right side such that the following two conditions are fulfilled. (1) The separators are nested, meaning that the left side of every separator S is contained in all the left sides of separators following S. (2) For each pair of separators, gluing the left side of the first and the right side of the second separator results in an r-outerplanar graph. Herein, gluing means to take the disjoint union and identify the vertices in the separators with the same labels. We apply the sequences as above to the problem of finding an r-outerplanar hypergraph support. That is, the problem is for a given hypergraph to find an r-outerplanar graph on the same vertex set such that each hyperedge induces a connected subgraph. We give an alternative proof that this problem is (strongly uniformly) fixed-parameter tractable with respect to r +m where m is the number of hyperedges in the hypergraph.
منابع مشابه
Hamilton cycles in plane triangulations
We extend Whitney’s Theorem that every plane triangulation without separating triangles is hamiltonian by allowing some separating triangles. More precisely we define a decomposition of a plane triangulation G into 4-connected ‘pieces’ and show that if each piece shares a triangle with at most three other pieces then G is hamiltonian. We provide an example to show that our hypothesis that ‘each...
متن کاملEdge covering pseudo-outerplanar graphs with forests
A graph is called pseudo-outerplanar if each block has an embedding on the plane in such a way that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another. In this paper, we prove that each pseudo-outerplanar graph admits edge decompositions into a linear forest and an outerplanar graph, or a star forest and an outerpla...
متن کاملMiscellaneous properties of embeddings of line, total and middle graphs
Chartrand et al. (J. Combin. Theory Ser. B 10 (1971) 12–41) proved that the line graph of a graph G is outerplanar if and only if the total graph of G is planar. In this paper, we prove that these two conditions are equivalent to the middle graph of G been generalized outerplanar. Also, we show that a total graph is generalized outerplanar if and only if it is outerplanar. Later on, we characte...
متن کاملLarge induced acyclic and outerplanar subgraphs of low-treewidth planar graphs
Albertson and Berman conjectured that every planar graph has an induced forest on half of its vertices. The best known lower bound, due to Borodin, is that every planar graph has an induced forest on two fifths of its vertices. In a related result, Chartran and Kronk, proved that the vertices of every planar graph can be partitioned into three sets, each of which induce a forest. We show tighte...
متن کاملList-colourings of Near-outerplanar Graphs
A list-colouring of a graph is an assignment of a colour to each vertex v from its own list L(v) of colours. Instead of colouring vertices we may want to colour other elements of a graph such as edges, faces, or any combination of vertices, edges and faces. In this thesis we will study several of these different types of list-colouring, each for the class of a near-outerplanar graphs. Since a g...
متن کامل