A note on quasi-robust cycle bases

We investigate here some aspects of cycle bases of undirected graphs that allow the iterative construction of all elementary cycles. We introduce the concept of quasi-robust bases as a generalization of the notion of robust bases and demonstrate that a certain class of bases of the complete bipartite graphs K m,n with m,n _> 5 is quasi-robust but not robust. We furthermore disprove a conjecture for cycle bases of Cartesian product graphs

Space-Efficient Biconnected Components and Recognition of Outerplanar Graphs

We present space-efficient algorithms for computing cut vertices in a given graph with $n$ vertices and $m$ edges in linear time using $O(n+\min\{m,n\log \log n\})$ bits. With the same time and using $O(n+m)$ bits, we can compute the biconnected components of a graph. We use this result to show an algorithm for the recognition of (maximal) outerplanar graphs in $O(n\log \log n)$ time using $O(n)$ bits

An Improved Algorithm for Finding Maximum Outerplanar Subgraphs

We study the NP-complete Maximum Outerplanar Subgraph problem. The previous best known approximation ratio for this problem is 2/3. We propose a new approximation algorithm which improves the ratio to 7/10

An Improved Fixed-Parameter Algorithm for One-Page Crossing Minimization

Book embedding is one of the most well-known graph drawing models and is extensively studied in the literature. The special case where the number of pages is one is of particular interest: an embedding in this case has a natural circular representation useful for visualization and graphs that can be embedded in one page without crossings form an important graph class, namely that of outerplanar graphs. In this paper, we consider the problem of minimizing the number of crossings in a one-page book embedding, which we call one-page crossing minimization. Here, we are given a graph G with n vertices together with a non-negative integer k and are asked whether G can be embedded into a single page with at most k crossings. Bannister and Eppstein (GD 2014) showed that this problem is fixed-parameter tractable. Their algorithm is derived through the application of Courcelle\u27s theorem (on graph properties definable in the monadic second-order logic of graphs) and runs in f(L)n time, where L = 2^{O(k^2)} is the length of the formula defining the property that the one-page crossing number is at most k and f is a computable function without any known upper bound expressible as an elementary function. We give an explicit dynamic programming algorithm with a drastically improved running time of 2^{O(k log k)}n

Schematic Representation of Large Biconnected Graphs

Suppose that a biconnected graph is given, consisting of a large component plus several other smaller components, each separated from the main component by a separation pair. We investigate the existence and the computation time of schematic representations of the structure of such a graph where the main component is drawn as a disk, the vertices that take part in separation pairs are points on the boundary of the disk, and the small components are placed outside the disk and are represented as non-intersecting lunes connecting their separation~pairs. We consider several drawing conventions for such schematic representations, according to different ways to account for the size of the small components. We map the problem of testing for the existence of such representations to the one of testing for the existence of suitably constrained $1$-page book-embeddings and propose several polynomial-time and pseudo-polynomial-time algorithms.Comment: Appears in the Proceedings of the 28th International Symposium on Graph Drawing and Network Visualization (GD 2020