6664703
The number of spanning trees of plane graphs with reflective symmetry
A plane graph is called symmetric if it is invariant under the reflection across some straight line (called symmetry axis). Let G be a symmetric plane graph. We prove that if there is no edge in G intersected by its symmetry axis then the number of spanning trees of G can be expressed in terms of the product the number of spanning trees of two smaller graphs. each of which has about half the number of vertices of G.
32475390
k-Cycle resonant graphs
Abstract A connected graph G is said to be k -cycle resonant if, for 1 ⩽ t ⩽ k , any t disjoint cycles in G are mutually resonant, that is, there is a perfect matching M of G such that each of the t cycles is an M -alternating cycle. In this paper, we at the first time introduce the concept of k -cycle resonant graphs, and investigate some properties of k -cycle resonant graphs. Some simple necessary and sufficient conditions for a graph to be k -cycle resonant are given. The construction of k -cycle resonant hexagonal systems are also characterized.
205056734
Hamiltonicity of a Type of Interchange Graphs
Abstract Let R(r1,…, rm) and S(s1,…, sn) be two nonnegative integral vectors, and let U (R,S) denote the class of all m×n matrices of 0's and 1's having row sum vector R and column sum vector S. We prove that if s1s2⋯sn1, the interchange graph G(R,S) of U (R,S) is hamiltonian.
23589102
Bounds of the longest directed cycle length for minimal strong digraphs
Abstract In this paper we present the upper and lower bounds of the longest directed cycle length for minimal strong digraphs in terms of the numbers of vertices and arcs. These bounds are both sharp. In addition, we give analogous results for minimal 2-edge connected graphs.
46002685
3-Restricted Edge Connectivity Of Vertex Transitive Graphs
5788058
Heredity of the index of convergence of the line digraph
Let D be a digraph, and let Ln(D) and k(D) denote the nth iterated line digraph of D and the index of convergence of D, respectively. We prove that if D contains neither sources nor sinks, then: (1) k(Ln(D))=k(D)=0 if every connected component of D is a directed cycle. (2) k(Ln(D))=k(D)+n if there exists at least one connected component of D that is not a directed cycle. Finally, we prove that if D has no sources or no sinks then k(D) ≤ k(L(D)) ≤ k(D)+1.
44874837
Z-transformation graphs of perfect matchings of plane bipartite graphs
Abstract Let G be a plane bipartite graph with at least two perfect matchings. The Z -transformation graph, Z F ( G ), of G with respect to a specific set F of faces is defined as a graph on the perfect matchings of G such that two perfect matchings M 1 and M 2 are adjacent provided M 1 and M 2 differ only in a cycle that is the boundary of a face in F . If F is the set of all interior faces, Z F ( G ) is the usual Z -transformation graph; If F contains all faces of G it is a novel graph and called the total Z -transformation graph. In this paper, we give some simple characterizations for the Z -transformation graphs to be connected by applying the above new idea. Furthermore, we show that the total Z -transformation graph of G is 2-connected if G is elementary; the total Z -transformation digraph of G is strongly connected if and only if G is elementary.
15887745
The Rotation Graphs of Perfect Matchings of Plane Bipartite Graphs
Abstract In the present paper, the minimal proper alternating cycle (MPAC) rotation graph R(G) of perfect matchings of a plane bipartite graph G is defined. We show that an MPAC rotation graph R(G) of G is a directed rooted tree, and thus extend such a result for generalized polyhex graphs to arbitrary plane bipartite graphs. As an immediate result, we describe a one-to-one correspondence between MPAC systems and perfect matchings in G.
2376990
Power-aware processor scheduling under average delay constraints
In this paper, voltage scaling strategies for scheduling aperiodic tasks under average delay constraints are studied. Dynamic voltage scaling in single processor systems is formulated as a constrained stochastic optimization problem for which the optimal solution can be obtained using a combination of Lagrange relaxation and the value iteration method. For multiprocessor systems, we present a two-phase approach. In the first phase, the speed settings and static workload distribution of the processors are optimized to minimize the total power dissipation. Dynamic voltage scaling techniques are then applied to each individual processor in the second phase. Both homogeneous and heterogeneous systems have been investigated. Based on queueing theory, the proposed algorithms guarantee conformity to the average delay constraint. Moreover, our simulation experiments have shown they are effective for minimizing power consumption.
32440433
The FORTRAN Parallel Transformer and its Programming
The Fortran Parallel Transformer (FPT) is a parallelization tool for Fortran-77 programs. It is used for the automatic parallelization of loops, program transformations, dependence analysis, performance tuning and code generation for various platforms. FPT is able to deal with GOTO's by restructuring ill-structured code using hammock graph transformations. In this way more parallelism becomes detectable. The X-window based Programming Environment, PEFPT, extends FPT with interactive dependence analysis, the iteration space graph, ISG, and guided loop optimization. FPT contains a PVM (Parallel Virtual Machine) code generator which converts the parallel loops into PVM master- and slave-code for a network of workstations. This includes job scheduling, synchronization and optimized data communication. The productivity gained is about a factor of 10 in programming time and a significant speedup of the execution.
34749580
Plane elementary bipartite graphs
A connected graph is called elementary if the union of all perfect matchings forms a connected subgraph. In this paper we mainly study various properties of plane elementary bipartite graphs so that many important results previously obtained for hexagonal systems are treated in a unified way. Firstly, we show that a plane bipartite graph G is elementary if and only if the boundary of each face (including the infinite face) is an alternating cycle with respect to some perfect matching of G. For a plane bipartite graph G all interior vertices of which are of the same degree, a stronger result is obtained; namely, G is elementary if and only if the boundary of the infinite face of G is an alternating cycle with respect to some perfect matching of G. Second, the concept of the Z-transformation graph Z(G) of a hexagonal system G (whose vertices represent the perfect matchings of G) is extended to a plane bipartite graph G and some results analogous to those for hexagonal systems are obtained. A peripheral face f of G is called reducible if the removal of the internal vertices and edges of the path that is the intersection of f and the exterior face of G results in a plane elementary bipartite graph. Thirdly, we obtain the reducible face decomposition for plane elementary bipartite graphs. Furthermore, sharp upper and lower bounds for the number of reducible faces are derived. Conversely, we can construct any plane elementary bipartite graphs by adding new peripheral faces one by one. As applications of this approach, we give simple construction methods for several types of plane elementary bipartite graphs G that contain a forcing edge (which belongs to exactly one perfect matching of G) and whose Z-transformation graphs Z(G) contain vertices of degree one.