Biconnected Outerplanar Graphs Research Articles

Approximation algorithms for maximum weighted internal spanning trees in regular graphs and subdivisions of graphs

Abstract Let $G$ be a vertex-weighted connected graph of $n$ vertices and let $T$ be a spanning tree of $G$. We call $T$ a maximum weighted internal spanning tree of $G$ if the sum of the weights of the internal vertices of $T$ is the maximum over all spanning trees of $G$. The maximum weighted internal spanning tree (MaxwIST) problem asks to find such a spanning tree $T$ of $G$. The problem is NP-hard. We give an $O(dn)$ time approximation algorithm for $d$-regular graphs of $n=|V|$ vertices that computes a spanning tree with total weight of the internal vertices is at least $\frac{\beta _{d}}{\beta _{d} +d-2} - \epsilon $ of the total weight of all the vertices of the graph for any $\epsilon>0$, where $\beta _{d} = (d-1)H_{d-1}$, and $H_{d-1} = \sum _{i=1}^{d-1} i^{-1}$ is the $(d-1)$th harmonic number. For every $d \geq 3$ and $n_{0} \geq 1$, we show the construction of a $d$-regular graph of at least $n_{0}$ vertices, such that for any of its spanning trees, $\frac{w(I)}{w(V)}\le \frac{d}{d+1}$ holds. We give an $O(dn)$ time approximation algorithm for subdivisions of $d$-regular graphs, where the ratio of the internal weight of the spanning tree with the total vertex weight of the graph is at least $\frac{d-1}{2d-3} - \epsilon $ for $\epsilon>0$. We extend our study to $x$-subdivisions of Hamiltonian and hypoHamiltonian graphs, where each edge of the original Hamiltonian or hypoHamiltonian graph has been subdivided at least $x$ times. For those two graph classes, we show that there exists a spanning tree with internal vertex weight at least $1-\frac{2}{x-1}$ of the total vertex weight of the graph. Furthermore, we give $O(n)$ time algorithm for $x$-subdivisions of biconnected outerplanar graphs and $4$-connected planar graphs to achieve the above bound.

Four-searchable biconnected outerplanar graphs

This paper deals with constructing obstruction sets for two subclasses of 4-searchable graphs. We first characterize the 4-searchable biconnected outerplanar graphs by listing all graphs that cannot be their minors; we then give a constructive characterization of such graphs. We also characterize the 4-searchable biconnected generalized wheel graphs by listing all graphs that cannot be their minors.

HV-planarity: Algorithms and complexity

An HV-graph is a planar graph with vertex-degree at most four such that each edge is labeled either H (horizontal) or V (vertical). The HV-planarity testing problem asks whether an HV-graph admits an HV-drawing, that is, a planar drawing such that each edge with label H is drawn as a horizontal segment and each edge with label V is drawn as a vertical segment. We prove that the HV-planarity testing problem is NP-complete even for graphs with vertex-degree at most three, which answers an open question posed by both Manuch et al. [30] and Durocher et al. [17]. On the positive side, we prove that the HV-planarity testing problem can be solved in polynomial-time for series-parallel graphs. This result significantly extends a previous result by Durocher et al. about HV-planarity testing of biconnected outerplanar graphs of maximum vertex-degree three.

Locating median paths on connected outerplanar graphs

AbstractIn this article, we study the median path problem without length restrictions on the class of connected outerplanar graphs, assuming that weights equal to 1 are assigned to the edges of a graph G, and nonnegative weights are associated to its vertices. We provide an $O(kn)$ time algorithm, where n is the number of vertices of G and k is the number of blocks in G. As a byproduct, when G is a biconnected outerplanar graph, we provide a linear time algorithm to find a median path between two fixed vertices of G without restrictions on the length. In the literature, we did not find polynomial time algorithms for this problem on such classes of graphs. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011

Characterizations of restricted pairs of planar graphs allowing simultaneous embedding with fixed edges

A set of planar graphs { G 1 ( V , E 1 ) , … , G k ( V , E k ) } admits a simultaneous embedding if they can be drawn on the same pointset P of order n in the Euclidean plane such that each point in P corresponds one-to-one to a vertex in V and each edge in E i does not cross any other edge in E i (except at endpoints) for i ∈ { 1 , … , k } . A fixed edge is an edge ( u , v ) that is drawn using the same simple curve for each graph G i whose edge set E i contains the edge ( u , v ) . We give a necessary and sufficient condition for two graphs whose union is homeomorphic to K 5 or K 3 , 3 to admit a simultaneous embedding with fixed edges ( SEFE). This allows us to characterize the class of planar graphs that always have a SEFE with any other planar graph. We also characterize the class of biconnected outerplanar graphs that always have a SEFE with any other outerplanar graph. In both cases, we provide O ( n 4 ) -time algorithms to compute a SEFE.

A decomposition for total‐coloring partial‐grids and list‐total‐coloring outerplanar graphs

AbstractThe total chromatic number χT(G) is the least number of colors sufficient to color the elements (vertices and edges) of a graph G in such a way that no incident or adjacent elements receive the same color. In the present work, we obtain two results on total‐coloring. First, we extend the set of partial‐grids classified with respect to the total‐chromatic number, by proving that every 8‐chordal partial‐grid of maximum degree 3 has total chromatic number 4. Second, we prove a result on list‐total‐coloring biconnected outerplanar graphs. If for each element x of a biconnected outerplanar graph G there exists a set Lx of colors such that |Luw| = max{deg(u) + 1, deg(w) + 1} for each edge uw and |Lv| = 7 − δdeg(v),3 − 2δdeg(v),2 (where δi,j = 1 if i = j and δi,j = 0 if i ≠ j) for each vertex v, then there is a total‐coloring π of graph G such that π(x) ∈ Lx for each element x of G. The technique used in these two results is a decomposition by a cutset of two adjacent vertices, whose properties are discussed in the article. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011

Colored Simultaneous Geometric Embeddings and Universal Pointsets

Universal pointsets can be used for visualizing multiple relationships on the same set of objects or for visualizing dynamic graph processes. In simultaneous geometric embeddings, the same point in the plane is used to represent the same object as a way to preserve the viewer’s mental map. In colored simultaneous embeddings this restriction is relaxed, by allowing a given object to map to a subset of points in the plane. Specifically, consider a set of graphs on the same set of n vertices partitioned into k colors. Finding a corresponding set of k-colored points in the plane such that each vertex is mapped to a point of the same color so as to allow a straight-line plane drawing of each graph is the problem of colored simultaneous geometric embedding. For n-vertex paths, we show that there exist universal pointsets of size n, colored with two or three colors. We use this result to construct colored simultaneous geometric embeddings for a 2-colored tree together with any number of 2-colored paths, and more generally, a 2-colored outerplanar graph together with any number of 2-colored paths. For n-vertex trees, we construct small near-universal pointsets for 3-colored caterpillars of size n, 3-colored radius-2 stars of size n+3, and 2-colored spiders of size n. For n-vertex outerplanar graphs, we show that these same universal pointsets also suffice for 3-colored K 3-caterpillars, 3-colored K 3-stars, and 2-colored fans, respectively. We also present several negative results, showing that there exist a 2-colored planar graph and pseudo-forest, three 3-colored outerplanar graphs, four 4-colored pseudo-forests, three 5-colored pseudo-forests, five 5-colored paths, two 6-colored biconnected outerplanar graphs, three 6-colored cycles, four 6-colored paths, and three 9-colored paths that cannot be simultaneously embedded.

Characterizing Simultaneous Embedding with Fixed Edges

Abstract A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V of n vertices in the plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same Jordan curve in the simultaneous drawing. We give a necessary condition for when pairs of graphs can have a simultaneous embedding with fixed edges ( SEFE ). This allows us to determine for which (outer)planar graphs always have a SEFE with all (outer)planar graphs with O ( n 2 lg n ) time drawing algorithms. This allows us to decide in O ( n ) time whether a pair of biconnected outerplanar graphs has a SEFE .

Pathwidth of outerplanar graphs

AbstractWe are interested in the relation between the pathwidth of a biconnected outerplanar graph and the pathwidth of its (geometric) dual. Bodlaender and Fomin [3], after having proved that the pathwidth of every biconnected outerplanar graph is always at most twice the pathwidth of its (geometric) dual plus two, conjectured that there exists a constant c such that the pathwidth of every biconnected outerplanar graph is at most c plus the pathwidth of its dual. They also conjectured that this was actually true with c being one for every biconnected planar graph. Fomin [10] proved that the second conjecture is true for all planar triangulations. First, we construct for each p ≥ 1, a biconnected outerplanar graph of pathwidth 2p + 1 whose (geometric) dual has pathwidth p + 1, thereby disproving both conjectures. Next, we also disprove two other conjectures (one of Bodlaender and Fomin [3], implied by one of Fomin [10]. Finally we prove, in an algorithmic way, that the pathwidth of every biconnected outerplanar graph is at most twice the pathwidth of its (geometric) dual minus one. A tight interval for the studied relation is therefore obtained, and we show that all cases in the interval happen. © 2006 Wiley Periodicals, Inc. J Graph Theory 55: 27–41, 2007

Approximation of pathwidth of outerplanar graphs

There exists a polynomial time algorithm to compute the pathwidth of outerplanar graphs, but the large exponent makes this algorithm impractical. In this paper, we give an algorithm that, given a biconnected outerplanar graph G, finds a path decomposition of G of pathwidth at most twice the pathwidth of G plus one. To obtain the result, several relations between the pathwidth of a biconnected outerplanar graph and its dual are established.

Maximum packing for biconnected outerplanar graphs

The problem of determining the maximum number of vertex-disjoint subgraphs of a biconnected outerplanar graph H on n h vertices isomorphic to a “pattern” biconnected outerplanar graph G on n g vertices is shown to be solvable in time O(( n h n g ) 2).

Augmenting Outerplanar Graphs

In this paper, we show that for outerplanar graphsGthe problem of augmentingGby adding a minimum number of edges such that the augmented graphG′ is planar and bridge-connected, biconnected, or triconnected can be solved in linear time and space. It is also shown that augmenting a biconnected outerplanar graph to a maximal outerplanar graph while minimizing the maximum degree can be achieved in polynomial time. These augmentation problems arise in the area of drawing outerplanar graphs.

Fast detection and display of symmetry in outerplanar graphs

The automatic construction of good drawings of abstract graphs is a problem of practical importance. Symmetry appears as one of the main criteria for achieving goodness. Algorithms are developed for enumerating all planar axial and rotational symmetrics of a biconnected outerplanar graph, and it is shown how to construct a drawing which simultaneously displays all these symmetries. These results are then expanded to obtain algorithms for the detection and display of outerplanar axial and rotational symmetry in the entire class of outerplanar graphs. All algorithms run in both time and space which are linear in the size of the graph, and hence are optimal to within a constant factor.

Subgraph isomorphism for biconnected outerplanar graphs in cubic time

A dynamic programming algorithm for determining whether a biconnected outerplanar graph is isomorphic to a subgraph of another biconnected outerplanar graph is presented. The algorithm runs in cubic time.