Class Of Outerplanar Graphs Research Articles

Preprocessing for Outerplanar Vertex Deletion: An Elementary Kernel of Quartic Size

In the {varvec{mathcal {F}}}-Minor-Free Deletion problem one is given an undirected graph {varvec{G}}, an integer {varvec{k}}, and the task is to determine whether there exists a vertex set {varvec{S}} of size at most {varvec{k}}, so that {varvec{G}}-{varvec{S}} contains no graph from the finite family {varvec{mathcal {F}}} as a minor. It is known that whenever {varvec{mathcal {F}}} contains at least one planar graph, then {varvec{mathcal {F}}}-Minor-Free Deletion admits a polynomial kernel, that is, there is a polynomial-time algorithm that outputs an equivalent instance of size {varvec{k}}^{{varvec{mathcal {O}}}{} {textbf {(1)}}} [Fomin, Lokshtanov, Misra, Saurabh; FOCS 2012]. However, this result relies on non-constructive arguments based on well-quasi-ordering and does not provide a concrete bound on the kernel size. We study the Outerplanar Deletion problem, in which we want to remove at most {varvec{k}} vertices from a graph to make it outerplanar. This is a special case of {varvec{mathcal {F}}}-Minor-Free Deletion for the family {varvec{mathcal {F}}} = {{varvec{K}}_{{textbf {4}}}, {varvec{K}}_{{{textbf {2,3}}}}}. The class of outerplanar graphs is arguably the simplest class of graphs for which no explicit kernelization size bounds are known. By exploiting the combinatorial properties of outerplanar graphs we present elementary reduction rules decreasing the size of a graph. This yields a constructive kernel with {varvec{mathcal {O}}}({varvec{k}}^{textbf {4}}) vertices and edges. As a corollary, we derive that any minor-minimal obstruction to having an outerplanar deletion set of size {varvec{k}} has {varvec{mathcal {O}}}({varvec{k}}^{textbf {4}}) vertices and edges.

The sandpile group of a polygon flower

Let Ct be a cycle of length t, and let P1,…,Pt be t polygon chains. A polygon flower F=(Ct;P1,…,Pt) is a graph obtained by identifying the ith edge of Ct with an edge ei that belongs to an end-polygon of Pi for i=1,…,t. In this paper, we first give an explicit formula for the sandpile group S(F) of F, which shows that the structure of S(F) only depends on the numbers of spanning trees of Pi and Pi∕ei, i=1,…,t. By analyzing the arithmetic properties of those numbers, we give a simple formula for the minimum number of generators of S(F), by which a sufficient and necessary condition for S(F) being cyclic is obtained. Finally, we obtain a classification of edges that generate the sandpile group.Although the main results concern only a class of outerplanar graphs, the proof methods used in the paper may be of much more general interest. We make use of the graph structure to find a set of generators and a relation matrix R, which has the same form for any F and has much smaller size than that of the (reduced) Laplacian matrix, which is the most popular relation matrix used to study the sandpile group of a graph.

Algorithms for Outerplanar Graph Roots and Graph Roots of Pathwidth at Most 2

Deciding if a graph has a square root is a classical problem, which has been studied extensively both from graph-theoretic and algorithmic perspective. As the problem is NP-complete, substantial effort has been dedicated to determining the complexity of deciding if a graph has a square root belonging to some specific graph class $${{\mathcal {H}}}$$ . There are both polynomial-time solvable and NP-complete results in this direction, depending on $${{\mathcal {H}}}$$ . We present a general framework for the problem if $$\mathcal{H}$$ is a class of sparse graphs. This enables us to generalize a number of known results and to give polynomial-time algorithms for the cases where $${{\mathcal {H}}}$$ is the class of outerplanar graphs and $${{\mathcal {H}}}$$ is the class of graphs of pathwidth at most 2.

Enumeration of labeled outerplanar bicyclic and tricyclic graphs

The class of outerplanar graphs is used for testing the average complexity of algorithms on graphs. A random labeled outerplanar graph can be generated by a polynomial algorithm based on the results of an enumeration of such graphs. By a bicyclic (tricyclic) graph we mean a connected graph with cyclomatic number 2 (respectively, 3). We find explicit formulas for the number of labeled connected outerplanar bicyclic and tricyclic graphs with n vertices and also obtain asymptotics for the number of these graphs for large n. Moreover, we obtain explicit formulas for the number of labeled outerplanar bicyclic and tricyclic n-vertex blocks and deduce the corresponding asymptotics for large n.

Scaling limits of random graphs from subcritical classes

We study the uniform random graph C-n with n vertices drawn from a subcritical class of connected graphs. Our main result is that the resealed graph C-n / root n converges to the Brownian continuum random tree T-e multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide sub-Gaussian tail bounds for the diameter D (C-n) and height H(C-n(center dot)) of the rooted random graph C-n(center dot) We give analytic expressions for the scaling factor in several cases, including for example the class of outerplanar graphs. Our methods also enable us to study first passage percolation on C-n, where we also show the convergence to T-e under an appropriate rescaling.

Excluded-Minor Characterization of Apex-Outerplanar Graphs

The class of outerplanar graphs is minor-closed and can be characterized by two excluded minors: $$K_4$$K4 and $$K_{2,3}$$K2,3. The class of graphs that contain a vertex whose removal leaves an outerplanar graph is also minor-closed. We provide the complete list of 57 excluded minors for this class.

Frequent subgraph mining in outerplanar graphs

In recent years there has been an increased interest in frequent pattern discovery in large databases of graph structured objects. While the frequent connected subgraph mining problem for tree datasets can be solved in incremental polynomial time, it becomes intractable for arbitrary graph databases. Existing approaches have therefore resorted to various heuristic strategies and restrictions of the search space, but have not identified a practically relevant tractable graph class beyond trees. In this paper, we consider the class of outerplanar graphs, a strict generalization of trees, develop a frequent subgraph mining algorithm for outerplanar graphs, and show that it works in incremental polynomial time for the practically relevant subclass of well-behaved outerplanar graphs, i.e., which have only polynomially many simple cycles. We evaluate the algorithm empirically on chemo- and bioinformatics applications.

On-line Ramsey Theory

The Ramsey game we consider in this paper is played on an unbounded set of vertices by two players, called Builder and Painter. In one move Builder introduces a new edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed target graph $H$, keeping the constructed graph in a prescribed class ${\cal G}$. The main problem is to recognize the winner for a given pair $H,{\cal G}$. In particular, we prove that Builder has a winning strategy for any $k$-colorable graph $H$ in the game played on $k$-colorable graphs. Another class of graphs with this strange self-unavoidability property is the class of forests. We show that the class of outerplanar graphs does not have this property. The question of whether planar graphs are self-unavoidable is left open. We also consider a multicolor version of Ramsey on-line game. To extend our main result for $3$-colorable graphs we introduce another Ramsey type game, which seems interesting in its own right.

Minimal reducible bounds for 1-nonouterplanar graphs

We consider a subclass of planar graphs, slightly wider than the class of outerplanar graphs. We prove that for this class of graphs there are infinitely many minimal reducible bounds in contrast with the class of outerplanar graphs, and describe all of them.

Defective colorings of graphs in surfaces: Partitions into subgraphs of bounded valency

AbstractWe call a graph (m, k)‐colorable if its vertices can be colored with m colors in such a way that each vertex is adjacent to at most k vertices of the same color as itself. For the class of planar graphs, and the class of outerplanar graphs, we determine all pairs (m, k) such that every graph in the class is (m, k)‐colorable. We include an elementary proof (not assuming the truth of the four‐color theorem) that every planar graph is (4, 1)‐colorable. Finally, we prove that, for each compact surface S, there is an integer k = k(S) such that every graph in S can be (4, k)‐colored; we conjecture that 4 can be replaced by 3 in this statement.