Abstract: We prove a theorem on paths with prescribed ends in a planar graph which extends Tutte's theorem on cycles in planar graphs [9] and implies the conjecture of Plummer [5] asserting that every 4-connected planar graph is Hamiltonian-connected.

Abstract: : Planar embedding with minimal area of graphs on an integer grid is an interesting problem in VLSI (Very Large Scale Integrated) theory. Valiant gave an algorithm to construct a planar embeddding for trees in linear area; he also proved that there are planar graphs that require quadratic area. We fill in a spectrum between Valiant's results by showing that an N-node planar graphs has a planar embedding with area 0(NF), where F is a bound on the path length from any node to the exterior face. In particular, an outerplanar graph can be embedded without crossovers in linear area. This bound is tight, up to constant factors: for any N and F, there exist graphs requiring omega(NF) area for planar embedding. Also, finding a minimal embedding area is shown to be Nu-complete for forests, and hence for more general types of graphs. (author)

Abstract: The paper deals with a subfamily of those planar graphs which have outerplanar intersection of their MacLane cycle basis. These graphs have been known as Halin graphs. Their connectivity properties, structure of cycles, and feasible embeddings in the plane are discussed here. This paper also presents some initial investigations of NP-complete problems restricted to the family of Halin graphs.

Abstract: Flow problems in planar networks are investigated. (i) Two algorithms for finding maximum flows in directed planar networks (hence in any planar network) are presented. The first is an O(n('3/2)log n) divide-and-conquer algorithm, and the second is an O(p n log n) algorithm, where p is the fewest number of faces to be crossed while going from the source to the sink in the embedding. (ii) A reduction from the planar circulation problem to the shortest-path problem is exhibited, which yields an O(n('3/2)) algorithm for finding a circulation in a planar network. This reduction also yields an O(n('3/2)) algorithm for constructing a flow (if feasible) of a given value in a planar network, even if non-zero lower bounds on the edges are allowed. (iii) On-line algorithms are given for updating flows and circulations, and reoptimizing maximum flows in planar networks, under certain changes in capacities and lower bounds. (iv) O(log('2)n) parallel algorithms are presented for finding min-cuts and maximum flows in undirected planar networks, constructing planar separators, and finding feasible circulations and flows in planar networks. (v) Some properties proven in the thesis also apply to general networks. For instance, a reduction is exhibited, which shows that the maximum-flow problem in general networks is equal in complexity to a seemingly different problem. (vi) Interesting properties of planar graphs are proven. For instance, a new linear-time characterization of the planar separator theorem is shown, in terms of mutually non-containing and non-intersecting closed Jordan Curves.

Abstract: This thesis introduces a new class of graphs called (DELTA) - Y reducible graphs and presents a linear time algorithm for the computation of the all-terminal reliability of a graph in this class. The algorithm is based on special properties of this class. These properties are derived from a general characterization of the class as developed in this thesis. The class of (DELTA) - Y reducible graphs includes all the classes of graphs for which polynomial network reliability algorithms have been known so far, namely the series-parallel graphs, wheel-graphs, ladder networks and graphs in normal form. More precisely, a graph is called (DELTA) - Y Reducible if it can be reduced to an edge by a recursive application of series, parallel and (DELTA) - Y replacements. It is shown that a nonseparable graph is (DELTA) - Y reducible if and only if it is a planar graph that contains no subgraph that is a subdivision of the cube or the decahedron, where the cube is the graph corresponding to the 3-dimensional cube and the decahedron is a suspension graph with seven vertices. To construct the O((VBAR)V(VBAR)) algorithm to compute the all terminal reliability of a (DELTA) - Y reducible graph G = (V,E), the following properties of G are derived and used. (a) A planar embedding of G has a face consisting of three edges, called a (DELTA) of G, such that the replacement of this face by a Y yields a graph that is again (DELTA) - Y reducible. Those (DELTA)s that have this property are characterized. (b) G contains a subgraph G' such that G' is attached to G - G' through three vertices and is a subgraph of one of six special structures. G' has less than ten vertices and is easily identifiable. Further the replacement of G' by a Y yields a graph that is again (DELTA) - Y reducible. To derive the characterization of the (DELTA) - Y reducible graphs, an important subclass is introduced and studied. This subclass, called the class of Inner-Four-Cycle-Free graphs, includes, as special cases, the series-parallel graphs and the graphs in normal form. A planar graph G = (V,E) is an Inner-Four-Cycle-Free graph (IFCF-graph) if after all series and parallel replacements it has a planar embedding G' with a face f such that any cycle of G' having four or more edges contains a vertex of f. An O((VBAR)V(VBAR)('2)) algorithm to compute the all-terminal reliability of any graph whose triconnected components are IFCF-graphs is presented. Finally, it is illustrated that the polynomial algorithms constructed for (DELTA) - Y reducible graphs can be combined with edge factoring in order to compute the reliability of a general graph more efficiently than any other method known so far.

Abstract: It is proved that any edge of a 4-connected non-planar graph G of order at least 6 lies in a subdivision of K3,3 in G. For any 3-connected non-planar graph G of order at least 6 we show that G contains at most four edges which belong to no subdivisions of K3,3 in G.

Abstract: The embedding of a planar graph G is geometrically determined for a given Tremaux tree of G , by the choice, left or right for each half-line of the cotree, of a side of embedding according to a tree-chain. A characterization of the circular order at each vertex allows an effective embedding in linear time. It provides a new outlook concerning automatic network-design, of electrical networks in particular.

Abstract: Publisher Summary This chapter discusses two kinds of graphs having certain topological properties. The chapter discusses the way all 4-connected planar graphs are edge-reconstructible. The chapter also discusses the edge-reconstruction of certain graphs that triangulate surfaces. There are two reconstruction techniques to establish the results. The conditions of 4-connectedness and planarity allow us to make heavy use of embedding properties and of Kuratowski's characterization of planar graphs. In general, these are no longer available for surfaces of higher genera.

Abstract: Call a connected planar graphG legal if it has at least two nodes, no parallel edges or self-loops and at most two terminals (degree 1 nodes) and all terminals and degree 2 nodes are exterior This class of graphs arose in connection with a two-dimensional generating system for modeling growth by binary cell division Showing that any permitted pattern can be generated properly requires a matching or pairing lemma The vertex set of a legal graph withn nodes can be split intop adjacent pairs ands singletons withs p, resulting in a matching which includes at least\(2\left[ {\frac{n}{3}} \right]\) nodes This bound is sharp in the sense that there are legal graphs for which this matching is maximum The matching can be implemented by a linear time algorithm A legal graph witht terminals and n≥4 nodes has a spanning tree with at most\(\left[ {\frac{{n - t}}{2}} \right] + t\) terminals; this bound is sharp Such a spanning tree can be constructed by an algorithm which operates in almost linear time

Abstract: Publisher Summary This chapter investigates the values N 2 (X, n, m) and N 3 (X, n, m) The chapter discusses theorems including vectors and graphs, and connections with probability theory If ξ and η are (independent and identically distributed) arbitrary random vectors then a ‘continuous' variant of Theorem 1—that is, of Turan's theorem, where “number of” (vertices, edges) is substituted by ‘measure of’ The problems concerning two-sums led to graph problems Similarly, the three-sums lead to three-graphs This makes these problems much harder because very little is known about three-graphs Some geometry to be able to use graph theory is required

Abstract: Let Fn,k, be the classes of maximal planar graphs Gn (without loops or multiple edges) having k vertices of degree 5 and n-k vertices of degree more than 5. Hakimi and Schmeichel proved that if Gn ɛ Fn,k and k=12, 13, then Gn is 5-connected. In this paper the author gives some general conditions in terms of the integers n and k for a graph Gn ɛ Fn,k to be p-connected, with p=4,5.

Abstract: Each square in a regular square lattice is given a colour according to a common probability distribution. Edges belonging to two squares of the same colour are deleted. The numbers of faces and components of the remaining sublattice are investigated.

Abstract: Planar graphs have recently been characterised as those which have no strict elegant odd ring of circuits. Here we generalise that result by showing that its dual yields a theorem that is valid for all graphs.

Abstract: We shall introduce the concepts of pseudoadjacent, pseudocoatraction and pseudoho­ momorphism. We shall show, with the help of these concepts, in theorem 2 that every quadrilateral graph V (which is a 4-vertexgraph) has chromatic number 2. From this theorem follows our main result in theorem 4 that every 2n+ 1-vertex free graph has chromatic number 2.

Abstract: Graph decomposition and data structures techniques are presented that make possible faster algorithms for shortest paths in planar graphs. Improved algorithms are presented for the single source problem, the all pairs problem, and the problem of finding a minimum cut in an undirected graph.

Abstract: The doubly indexed Whitney numbers of a finite, ranked partially ordered set L are (the first kind) w,, -- E{(x', xJ): x', xl E L with ranks i, j} and (the second kind) WJ the number of (x', x') with x' < xJ. When L has a 0 element, the ordinary (simply indexed) Whitney numbers are w, = wo, and W, = Wo, - W,,. Building on work of Stanley and Zaslavsky we show how to interpret the magnitudes of Whitney numbers of geometric lattices and semilattices arising in geometry and graph theory. For example: The number of regions, or of k-dimen- sional faces for any k, of an arrangement of hyperplanes in real projective or affine space, that do not meet an arbitrary hyperplane in general position. The number of vertices of a zonotope P inside the visible boundary as seen from a distant point on a generating line of P. The number of non-Radon partitions of a Euclidean point set not due to a separating hyperplane through a fixed point. The number of acyclic orientations of a graph (Stanley's theorem, with a new, geometrical proof); the number with a fixed unique source; the number whose set of increasing arcs (in a fixed ordering of the vertices) has exactly q sources (generalizing Renyi's enumera- tion of permutations with q "outstanding" elements). The number of totally cyclic orientatiofis of a plane graph in which there is no clockwise directed cycle. The number of acyclic orientations of a signed graph satisfying conditions analogous to an unsigned graph's having a unique source.

Abstract: A Halin graphH=T∪C is obtained by embedding a treeT having no nodes of degree 2 in the plane, and then adding a cycleC to join the leaves ofT in such a way that the resulting graph is planar. These graphs are edge minimal 3-connected, hamiltonian, and in general have large numbers of hamilton cycles. We show that for arbitrary real edge costs the travelling salesman problem can be polynomially solved for such a graph, and we give an explicit linear description of the travelling salesman polytope (the convex hull of the incidence vectors of the hamilton cycles) for such a graph.

Abstract: Let N be a planar undirected network with distinguished vertices s, t, a total of n vertices, and each edge labeled with a positive real (the edge’s cost) from a set L. This paper presents an algorithm for computing a minimum (cost) s-t cut of N. For general L, this algorithm runs in time $O(n\log ^2 (n))$. For the case when L contains only integers$ \leqq n^{O(1)} $, the algorithm runs in time $O(n\log (n)\log \log (n))$. Our algorithm also constructs a minimum s-t cut of a planar graph (i.e., for the case $L = \{ 1\} $) in time $O(n\log (n))$. Our algorithm can also be used to compute a minimum cut for a general undirected planar network.The fastest previous algorithm for computing a minimum s-t cut of a planar undirected network (Itai and Shiloach [SIAM J. Comput., 8 (1979), pp. 135–150]) has time $O(n^2 \log (n))$; the s-t cut is a byproduct of the maximum flow computed by their algorithm. The best previous time bound for minimum s-t cut of a planar graph (Cheston, Probert and Saxton [report, Dept. Co...