Solving puzzles by O. Ore's method

2020 ◽  
pp. 22-30
Author(s):  
Abdulkarim Magomedov ◽  
S.A. Lawrencenko
Keyword(s):  
Graph Theory ◽  
Directed Path ◽  
Acyclic Digraph ◽  
Different Origins

In some cases, the formalisation of the puzzle in terms of graph theory allows us to solve the puzzle by finding a path in a connected acyclic digraph. We follow the approach taken in Ore's book on graph theory. In the present paper we demonstrate the approach on problems of different origins. In each case, the problem is restated in terms of a connected acyclic digraph whose nodes are certain states and whose directed arcs are transitions between states; then it is shown how to reduce the problem to finding a directed path between the nodes of the constructed digraph.

scholarly journals Controllability of Weighted and Directed Networks with Nonidentical Node Dynamics

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Linying Xiang ◽  
Jonathan J. H. Zhu ◽  
Fei Chen ◽  
Guanrong Chen
Keyword(s):  
Graph Theory ◽  
Control Theory ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Directed Path ◽  
Directed Networks ◽  
Leaders And Followers ◽  
Heterogenous Network

The concept of controllability from control theory is applied to weighted and directed networks with heterogenous linear or linearized node dynamics subject to exogenous inputs, where the nodes are grouped into leaders and followers. Under this framework, the controllability of the controlled network can be decomposed into two independent problems: the controllability of the isolated leader subsystem and the controllability of the extended follower subsystem. Some necessary and/or sufficient conditions for the controllability of the leader-follower network are derived based on matrix theory and graph theory. In particular, it is shown that a single-leader network is controllable if it is a directed path or cycle, but it is uncontrollable for a complete digraph or a star digraph in general. Furthermore, some approaches to improving the controllability of a heterogenous network are presented. Some simulation examples are given for illustration and verification.

scholarly journals p-box: a new graph model

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Mauricio Soto ◽  
Christopher Thraves-Caro
Keyword(s):  
Graph Theory ◽  
Graph Model ◽  
Directed Path ◽  
Outerplanar Graphs ◽  
New Model ◽  
Model Based ◽  
Representative Element ◽  
International Audience ◽  
Graph Families

Graph Theory International audience In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.

On the shape of a random acyclic digraph

1989 ◽  
Vol 106 (3) ◽  
pp. 459-465 ◽  
Author(s):  
Brendan D. Mckay
Keyword(s):  
Directed Path ◽  
Acyclic Digraph ◽  
Normally Distributed

AbstractIf D is an acyclic digraph, define the height h = h(D) to be the length of the longest directed path in D. We prove that the values of h(D) over all labelled acyclic digraphs D on n vertices are asymptotically normally distributed with mean Cn and variance C′n, where C ≈ 0·764334 and C′ ≈ 0·145210. Furthermore, define V0(D) to be the set of sinks (vertices of out-degree 0) and, for r ≥ 1, define Vr(D) to be the set of vertices ν such that the longest directed path from ν to V0(D) has length r. For each k ≥ 1, let nk(D) be the number of sets Vt(D) which have size k. We prove that, for fixed k, the values of nk(D) over all labelled acyclic digraphs D on n vertices are asymptotically normally distributed with mean Ckn and variance C′kn, for positive constants Ck and C′k. Results of Bender and Robinson imply that our claim holds also for unlabelled acyclic digraphs.

scholarly journals Extension of Gyárfás-Sumner Conjecture to Digraphs

10.37236/9906 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Pierre Aboulker ◽  
Pierre Charbit ◽  
Reza Naserasr
Keyword(s):  
Chromatic Number ◽  
Directed Path ◽  
Complete Multipartite Graph ◽  
Color Class ◽  
Acyclic Digraph ◽  
Minimum Number ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Dichromatic Number

The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices  in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has become the focus of numerous works. In this work we look at possible extensions of the Gyárfás-Sumner conjecture. In particular, we conjecture a simple characterization  of sets $\mathcal F$ of three digraphs such that every digraph with sufficiently large dichromatic number must contain a member of $\mathcal F$ as an induced subdigraph.  Among notable results, we prove that oriented $K_4$-free graphs without a directed path of length $3$ have bounded dichromatic number where a bound of $414$ is provided. We also show that an orientation of a complete multipartite graph with no directed triangle is $2$-colorable. To prove these results we introduce the notion of nice sets that might be of independent interest.

Anisotropy of damage productions in electron irradiated molybdenum

1978 ◽  
Vol 36 (1) ◽  
pp. 354-355
Author(s):  
K. Izui ◽  
S. Furuno ◽  
H. Otsu ◽  
T. Nishida ◽  
H. Maeta
Keyword(s):  
Electron Flux ◽  
Threshold Energy ◽  
High Energy ◽  
Voltage Electron ◽  
High Voltage Electron Microscope ◽  
Displacement Threshold ◽  
The Mean ◽  
Channeling Effect ◽  
Different Origins ◽  
Threshold Energies

Anisotropy of damage productions in crystals due to high energy electron bombardment are caused from two different origins. One is an anisotropic displacement threshold energy, and the other is an anisotropic distribution of electron flux near the atomic rows in crystals due to the electron channeling effect. By the n-beam dynamical calculations for germanium and molybdenum we have shown that electron flux at the atomic positions are from ∽4 to ∽7 times larger than the mean incident flux for the principal zone axis directions of incident 1 MeV electron beams, and concluded that such a locally increased electron flux results in an enhanced damage production. The present paper reports the experimental evidence for the enhanced damage production due to the locally increased electron flux and also the results of measurements of the displacement threshold energies for the <100>,<110> and <111> directions in molybdenum crystals by using a high voltage electron microscope.

Graph Theory, Coding Theory and Block Designs

1975 ◽  
Author(s):  
P. J. Cameron ◽  
J. H. van Lint
Keyword(s):  
Graph Theory ◽  
Coding Theory ◽  
Block Designs
Sign in / Sign up

Export Citation Format

Share Document