Space Complexity of the Directed Reachability Problem over Surface-Embedded Graphs

The graph reachability problem, the computational task of deciding whether there is a path between two given nodes in a graph is the canonical problem for studying space bounded computations. Three central open questions regarding the space complexity of the reachabil-ity problem over directed graphs are: (1) improving Savitch’s O(log2 n) space bound, (2) designing a polynomial-time algorithm with O(n1−) space bound, and (3) designing an unambiguous non-deterministic log-space (UL) algorithm. These are well-known open questions in complex-ity theory, and solving any one of them will be a major breakthrough. We will discuss some of the recent progress reported on these questions for certain subclasses of surface-embedded directed graphs

Topological drawings of complete bipartite graphs

Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. We consider a natural class of simple topological drawings of complete bipartite graphs, in which we require that one side of the vertex set bipartition lies on the outer boundary of the drawing. We investigate the combinatorics of such drawings. For this purpose, we define combinatorial encodings of the drawings by enumerating the distinct drawings of subgraphs isomorphic to K2,2 and K3,2, and investigate the constraints they must satisfy. We prove in particular that for complete bipartite graphs of the form K2,n and K3,n, such an encoding corresponds to a drawing if and only if it obeys consistency conditions on triples and quadruples. In the general case of Kk,n with k ≥ 2, we completely characterize and enumerate drawings in which the order of the edges around each vertex is the same for vertices on the same side of the bipartition.SCOPUS: cp.kinfo:eu-repo/semantics/publishe

LNCS

Simple drawings of graphs are those in which each pair of edges share at most one point, either a common endpoint or a proper crossing. In this paper we study the problem of extending a simple drawing D(G) of a graph G by inserting a set of edges from the complement of G into D(G) such that the result is a simple drawing. In the context of rectilinear drawings, the problem is trivial. For pseudolinear drawings, the existence of such an extension follows from Levi’s enlargement lemma. In contrast, we prove that deciding if a given set of edges can be inserted into a simple drawing is NP-complete. Moreover, we show that the maximization version of the problem is APX-hard. We also present a polynomial-time algorithm for deciding whether one edge uv can be inserted into D(G) when {u,v} is a dominating set for the graph G

Hamiltonian Alternating Paths on Bicolored Double-Chains

We find arbitrarily large finite sets S of points in general position in the plane with the following property. If the points of S are equitably 2-colored (i.e., the sizes of the two color classes differ by at most one), then there is a polygonal line consisting of straight-line segments with endpoints in S , which is Hamiltonian, non-crossing, and alternating (i.e., each point of S is visited exactly once, every two non-consecutive segments are disjoint, and every segment connects points of different colors). We show that the above property holds for so-called double-chains with each of the two chains containing at least one fifth of all the points. Our proof is constructive and can be turned into a linear-time algorithm. On the other hand, we show that the above property does not hold for double-chains in which one of the chains contains at most ≈ 1/29 of all the points

Inserting One Edge into a Simple Drawing Is Hard

A simple drawing D(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of G+e extending D(G). As a result of Levi’s Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is NP -complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles A and a pseudosegment σ , it can be decided in polynomial time whether there exists a pseudocircle Φσ extending σ for which A∪{Φσ} is again an arrangement of pseudocircles

LNCS

A simple drawing D(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of G+e extending D(G). As a result of Levi’s Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is NP -complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles A and a pseudosegment σ , it can be decided in polynomial time whether there exists a pseudocircle Φσ extending σ for which A∪{Φσ} is again an arrangement of pseudocircles