An infinite family of simple graphs underlying chiral, orientable reflexible and non-orientable rotary maps

The problem of finding a fixed point of a nonexpansive mapping on a hypercube is that it has a polynomial time algorithm. In fact, it is known that one can find a 2-satisfiability characterization of the set of all fixed points in polynomial time. This implies that the problem of finding a vertex that is a common fixed point of several given nonexpansive mappings on a hypercube is that it has a polynomial time algorithm. We consider the problem of finding a vertex that is a common fixed point of several given nonexpansive mappings on a more general Cartesian product of graphs. For a single nonexpansive mapping, a known polynomial time algorithm finds a fixed point and a 2-satisfiability-like characterization of all fixed points. We introduce graphs with a farthest point property (also called apiculate graphs in [H. J. Bandelt and V. Chepoi, The Algebra of Metric Betweenness: Subdirect Representations, Retracts, and Axiomatics, manuscript]), and show that finding a common fixed point of several nonexpansive mappings on Cartesian products of such graphs involves using a polynomial time algorithm. We generalize this result to any family of graphs having a majority function. By contrast, the smallest graph (in the sense of having the fewest vertices, and the fewest edges of those having the fewest vertices) without the farthest point property is K2,3 , and finding a vertex that is a fixed point of two given nonexpansive mappings (retractions) on a Cartesian product of graphs isomorphic to K2,3 is NP-complete. More generally, we exhibit an infinite family of graphs without the farthest point property giving NP-completeness. We show that for any family of graphs not having a majority function, the existence of a common fixed point of two nonexpansive mappings on Cartesian products of such graphs is NP-complete. This proves a dichotomy for the problem based on the existence of a majority function; a similar dichotomy is obtained for the special case of nonexpansive mappings that are retractions. Finally we characterize the families of chordal graphs corresponding to both dichotomies.

It is shown that every sufficiently large almost-5-connected non-planar graph contains a minor isomorphic to an arbitrarily large graph from one of six families of graphs. The graphs in these families are also almost-5-connected, by which we mean that they are 4-connected and all 4-separations contain a “small” side. As a corollary, every sufficiently large almost-5-connected non-planar graph contains both a K3, 4-minor and a -minor. The connectivity condition cannot be reduced to 4-connectivity, as there are known infinite families of 4-connected non-planar graphs that do not contain a K3, 4-minor. Similarly, there are known infinite families of 4-connected non-planar graphs that do not contain a -minor.

This paper introduces a family of simple bipartite graphs denoted by , named and constructed as polygon graphs. These polygon graphs are bicubic simple graphs possessing Hamiltonian cycles. Some important results are proved for these graphs. The girth of these graphs is counted as 6(). Polygon graph is isomorphic to a famous Pappus graph. Since polygon graphs are bipartite, therefore they can be used as Tanner graphs for low density parity check codes. This family of graphs may also be used in the design of efficient computer networks. Key words: Polygon graph, girth, Hamiltonian cycle, bicubic graph.

Given a simple graph G, an ordered pair (π, cπ) is said to be a gap- [k]-edge-labelling (resp. gap-[k]-vertex-labelling) ofG ifπ is an edge-labelling (vertex-labelling) on the set {1, . . . , k}, and cπ is a proper vertex-colouring such that every vertex of degree at least two has its colour induced by the largest difference among the labels of its incident edges (neighbours). The decision problems associated with these labellings are NP-complete for k ≥ 3, and even when k = 2 for some classes of graphs. This thesis presents a study of the computational complexity of these problems, structural properties for certain families of graphs and several labelling algorithms and techniques. First, we present an NP-completeness result for the family of subcubic bipartite graphs. Second, we present polynomial-time algorithms for families ofgraphs. Third, we introduce a new parameter associated with gap-[k]-vertex-labellings ofgraphs.

Let G = (V(G), E(G)) be a simple, finite and undirected graph of order n. Given a bijection f: V(G) ∪ E(G) → Zk such that for each edge uv ∈ E(G), f(u) + f(v) + f(uv) is constant C( mod k). Let nf(i) be the number of vertices and edges labeled by i under f. If |nf(i) - nf(j)| ≤ 1 for all 0 ≤ i < j ≤ k - 1, we say f is a k-totally magic cordial (k-TMC) labeling of G. A graph is said to be k-totally cordial magic if it admits a k-TMC labeling. In this paper, we give some ways to construct new families of k-TMC graphs from a known k-totally cordial magic graphs. We also give a sufficient condition for an odd graph to admit no k-TMC labeling. As a by-product, we determine the k-totally magic cordiality of many families of graphs.

The Tower of Hanoi problem is generalized by placing pegs on the vertices of a given directed graph G with two distinguished vertices, S and D , and allowing moves only along arcs of this graph. An optimal solution for such a graph G is an algorithm that completes the task of moving a tower of any given number of disks from S to D in a minimal number of disk moves. In this article we present an algorithm which solves the problem for two infinite families of graphs, and prove its optimality. To the best of our knowledge, this is the first optimality proof for an infinite family of graphs. Furthermore, we present a unified algorithm that solves the problem for a wider family of graphs and conjecture its optimality.

Let \(\mathrm{\Pi }\) be a family of graphs. In the classical \(\mathrm{\Pi }\)-Vertex Deletion problem, given a graph G and a positive integer k, the objective is to check whether there exists a subset S of at most k vertices such that \(G-S\) is in \(\mathrm{\Pi }\). In this paper, we introduce the conflict free version of this classical problem, namely Conflict Free \(\mathrm{\Pi }\)-Vertex Deletion (CF-\(\mathrm{\Pi }\)-VD), and study these problems from the viewpoint of classical and parameterized complexity. In the CF-\(\mathrm{\Pi }\)-VD problem, given two graphs G and H on the same vertex set and a positive integer k, the objective is to determine whether there exists a set \(S\subseteq V(G)\), of size at most k, such that \(G-S\) is in \(\mathrm{\Pi }\) and H[S] is edgeless. Initiating a systematic study of these problems is one of the main conceptual contribution of this work. We obtain several results on the conflict free version of several classical problems. Our first result shows that if \(\mathrm{\Pi }\) is characterized by a finite family of forbidden induced subgraphs then CF-\(\mathrm{\Pi }\)-VD is Fixed Parameter Tractable (FPT). Furthermore, we obtain improved algorithms for conflict free version of several well studied problems. Next, we show that if \(\mathrm{\Pi }\) is characterized by a “well-behaved” infinite family of forbidden induced subgraphs, then CF-\(\mathrm{\Pi }\)-VD is W[1]-hard. Motivated by this hardness result, we consider the parameterized complexity of CF-\(\mathrm{\Pi }\)-VD when H is restricted to well studied families of graphs. In particular, we show that the conflict free versions of several well-known problems such as Feedback Vertex Set, Odd Cycle Transversal, Chordal Vertex Deletion and Interval Vertex Deletion are FPT when H belongs to the families of d-degenerate graphs and nowhere dense graphs.

Let π be a family of graphs. In the classical π-Vertex Deletion problem, given a graph G and a positive integer k, the objective is to check whether there exists a subset S of at most k vertices such that G − S is in π. In this paper, we introduce the conflict free version of this classical problem, namely Conflict Free π-Vertex Deletion (CF-π-VD), and study this problem from the viewpoint of classical and parameterized complexity. In the CF-π-VD problem, given two graphs G and H on the same vertex set and a positive integer k, the objective is to determine whether there exists a set $S\subseteq V(G)$ , of size at most k, such that G − S is in π and H[S] is edgeless. Initiating a systematic study of these problems is one of the main conceptual contribution of this work. We obtain several results on the conflict free versions of several classical problems. Our first result shows that if π is characterized by a finite family of forbidden induced subgraphs then CF-π-VD is Fixed Parameter Tractable (FPT). Furthermore, we obtain improved algorithms for conflict free versions of several well studied problems. Next, we show that if π is characterized by a “well-behaved” infinite family of forbidden induced subgraphs, then CF-π-VD is W[1]-hard. Motivated by this hardness result, we consider the parameterized complexity of CF-π-VD when H is restricted to well studied families of graphs. In particular, we show that the conflict free version of several well-known problems such as Feedback Vertex Set, Odd Cycle Transversal, Chordal Vertex Deletion and Interval Vertex Deletion are FPT when H belongs to the families of d-degenerate graphs and nowhere dense graphs.

In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n, n − 1 , and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, a method is presented to locate fault-tolerant resolving sets by using classical resolving sets in graphs. The second part of the paper applies the proposed method to three infinite families of regular graphs and locates certain fault-tolerant resolving sets. By accumulating the obtained results with some known results in the literature, we present certain lower and upper bounds on the fault-tolerant metric dimension of these families of graphs. As a byproduct, it is shown that these families of graphs preserve a constant fault-tolerant resolvability structure.

Chromatic number via Turán number

Representing graph families with edge grammars

Results about the total chromatic number and the conformability of some families of circulant graphs

The paper is devoted to the implementations of the public key algorithms based on simple algebraic graphs A(n, K) and D(n, K) defined over the same finite commutative ring K. If K is a finite field both families are families of graphs with large cycle indicator. In fact, the family D(n, F q ) is a family of graphs of large girth (f.g.l.g.) with c = 1, their connected components CD(n, F q ) form the f.g.l.g. with the speed of growth 4/3. Family A(n, q), char F q ≠ 2 is a family of connected graphs with large cycle indicator with the largest possible speed of growth. The computer simulation demonstrates the advantage (better density which is the number of monomial expressions) of public rules derived from A(n, q) in comparison with symbolic algorithm based on graphs D(n, q).

The family of graphs is defined by taking the simple whee graph with rim vertices and then adding three extra vertices on every rim edge of the wheel. In this paper, the critical group of this whole family of graphs is investigated.

At this time, technology is developing very quickly and is increasingly sophisticated. This technological development is certainly closely related to the development of computer technology. A computer is able to control a series of electronic devices using an IC chip that can be filled with programs and logic called microprocessor technology. A microprocessor is a digital component of the VLSI (Very Large Scale Integration) type with very high circuit complexity that is capable of carrying out the functions of a CPU (Central Processing Unit). Among many applications, the problem of crossing number very interesting and important because of its application in the optimization of chip are required in a circuit layout of VLSI. Crossing number used to obtain the lower bound on the amount of chip area of VLSI devices like microprocessor and memory chips additionally, crossings in the circuit layout could cause short circuit and therefore worth minimized independent of the chip area consideration. Some graph can be seen as built by small pieces. A principal tool used in construction of crossing-critical graphs are tiles. In the tile concept, tiles can be arranged by gluing one tile to another in a linear or circular fashion. The series of tiles with circular fashion form an infinite graph family. In this way, the intersection number of this family of graphs can be determined. In this research, has been formed an infinite family graphs The graph formed by gluing together many copies of the tile in circular fashion, where the tile consists of identical tile sections. The results obtained show that the graph has 3-crossing-critical of a graph.