Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces

The Subgraph Isomorphism problem asks, given a host graph G on n vertices and a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P. The restriction of this problem to planar graphs has often been considered. After a sequence of improvements, the current best algorithm for planar graphs is a linear time algorithm by Dorn (STACS '10), with complexity $2^{O(k)} O(n)$. We generalize this result, by giving an algorithm of the same complexity for graphs that can be embedded in surfaces of bounded genus. At the same time, we simplify the algorithm and analysis. The key to these improvements is the introduction of surface split decompositions for bounded genus graphs, which generalize sphere cut decompositions for planar graphs. We extend the algorithm for the problem of counting and generating all subgraphs isomorphic to P, even for the case where P is disconnected. This answers an open question by Eppstein (SODA '95 / JGAA '99)

Defective and Clustered Graph Colouring

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A colouring has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter, graphs with given circumference, graphs excluding a fixed graph as an immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in the Electronic Journal of Combinatoric

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

From Graph Coloring to Receptor Clustering

1. Hued colorings for planar graphs, graphs of higher genus and K4-minor free graphs.;For integers k, r \u3e 0, a (k,r) -coloring of a graph G is a proper coloring of the vertices of G with k colors such that every vertex v of degree d(v) is adjacent to vertices with at least min{lcub}d(v) ,r{rcub} different colors. The r-hued chromatic number, denoted by Xr (G), is the smallest integer k for which a graph G has a ( k,r)-coloring. A list assignment L of G is a function that assigns to every vertex v of G a set L(v) of positive integers. For a given list assignment L of G, an ( L,r)-coloring of G is a proper coloring c of the vertices such that every vertex v of degree d(v) is adjacent to vertices with at least min{lcub} d(v),r{rcub} different colors and c(v) epsilon L(v). The r-hued choice number of G, XL,r(G), is the least integer k such that every list assignment L with | L(v)| = k, ∀ v epsilon V(G), permits an (L,r)-coloring. It is known that for any graph G, Xr(G) ≤ XL,r( G). Using Euler distributions, we proved the following results, where (ii) and (iii) are best possible. (i) If G is planar, then XL,2(G) ≤ 6. Moreover, XL,2( (G) ≤ 5 when Delta (G) ≤ 4. (ii) If G is planar, then X2( G) ≤ 5. (iii) If G is a graph with genus g(G) ≥ 1, then XL,2 (G) ≤ ½ 7+1+48gG .;Let K(r) = r + 3 if 2 ≤ r ≤ 3, and K(r) = 3r/2+1 if r≥ 4. We proved that if G is a K4-minor free graph, then (i) Xr(G) ≤ K(r), and the bound can be attained; (ii) XL,r(G) ≤ K( r)+1. This extends a previous result in [Discrete Math. 269 (2003) 303--309].;2. Quantitative description and impact of VEGF receptor clustering .;Cell membrane-bound receptors control signal initiation in many important cellular signaling pathways. Microscopic imaging and modern labeling techniques reveal that certain receptor types tend to co-localize in clusters, ranging from a few to hundreds of members. Here, we further develop a method of defining receptor clusters in the membrane based on their mutual distance, and apply it to a set of transmission microscopy (TEM) images of vascular endothelial growth factor (VEGF) receptors. We clarify the difference between the observed distributions and random placement. Moreover, we outline a model of clustering based on the hypothesis of pre-existing domains that have a high affinity for receptors. The observed results are consistent with the combination of two distributions, one corresponding to the placement of clusters, and the other to that of random placement of individual receptors within the clusters. Further, we use the preexisting domain model to calculate the probability distribution of cluster sizes. By comparing to the experimental result, we estimate the likely area and attractiveness of the clustering domains.;Furthermore, as VEGF signaling is involved in the process of blood vessel development and maintenance, it is of our interest to investigate the impact of VEGF receptors (VEGFR) clustering. VEGF signaling is initiated by binding of the bivalent VEGF ligand to the membrane-bound receptors (VEGFR), which in turn stimulates receptor dimerization. To address these questions, we have formulated the simplest possible model. We have postulated the existence of a single high affinity region in the cell membrane, which acts as a transient trap for receptors. We have defined an ODE model by introducing high- and low-density receptor variables and introduce the corresponding reactions from a realistic model of VEGF signal initiation. Finally, we use the model to investigate the relation between the degree of VEGFR concentration, ligand availability, and signaling. In conclusion, our simulation results provide a deeper understanding of the role of receptor clustering in cell signaling

Dynamic Programming for Graphs on Surfaces

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2^{O(k log k)} n steps. Our approach combines tools from topological graph theory and analytic combinatorics. In particular, we introduce a new type of branch decomposition called "surface cut decomposition", generalizing sphere cut decompositions of planar graphs introduced by Seymour and Thomas, which has nice combinatorial properties. Namely, the number of partial solutions that can be arranged on a surface cut decomposition can be upper-bounded by the number of non-crossing partitions on surfaces with boundary. It follows that partial solutions can be represented by a single-exponential (in the branchwidth k) number of configurations. This proves that, when applied on surface cut decompositions, dynamic programming runs in 2^{O(k)} n steps. That way, we considerably extend the class of problems that can be solved in running times with a single-exponential dependence on branchwidth and unify/improve most previous results in this direction.Comment: 28 pages, 3 figure