Network reliability with node failures

In this paper, we obtain three general reduction formulas to determine the orientable and nonorientable genera for complete tripartite graphs. As corollaries, we (1) reduce the determination of the orientable (nonorientable, respectively) genera of 75 percent (85 percent, respectively) of nonsymmetric (with respect to l,m, and n) Kl,m,n to that of Km,m,n, and (2) determine the orientable and nonorientable genera for several classes of complete tripartite graphs.

For a graph G and for non-negative integers p , q and r , the triplet ( p , q , r ) is said to be an admissible triplet, if 3 p + 4 q + 6 r = | E ( G ) | . If G admits a decomposition into p cycles of length 3 , q cycles of length 4 , and r cycles of length 6 for every admissible triplet ( p , q , r ) , then we say that G has a { C p 3 , C q 4 , C r 6 } -decomposition. In this paper, the necessary conditions for the existence of { C p 3 , C q 4 , C r 6 } -decomposition of K ℓ , m , n ( ℓ ≤ m ≤ n ) are proved to be sufficient. This affirmatively answers the problem raised in \emph{Decomposing complete tripartite graphs into cycles of lengths 3 and 4 , Discrete Math. 197/198 (1999), 123-135}. As a corollary, we deduce the main results of \emph{Decomposing complete tripartite graphs into cycles of lengths 3 and 4 , Discrete Math., 197/198, 123-135 (1999)} and \emph{Decompositions of complete tripartite graphs into cycles of lengths 3 and 6 , Austral. J. Combin., 73(1), 220-241 (2019)}.

Let G be a complete tripartite graph with n vertices. Its vertex set is U∪V∪W , where | U |=1 , | V |=s , | W |=t , and this graph is denoted as K 1,s,t . Let T 1,s,t ω denote the path-tree corresponding to the vertex ω∈U∪V∪W . In this paper, we mainly present the matching polynomial of the path-tree T 1,s,t ω corresponding to the complete tripartite graph.

Let X be a graph on n vertices and let B = {P (x) : x ∈ V (X)} be a collection of n subgraphs of X, one for each vertex, B is an orthogonal double cover (ODC) of X if every edge of X occurs in exactly two members of B and any two members share an edge whenever the corresponding vertices are adjacent in X and share no edges whenever the corresponding vertices are nonadjacent in X.The main question is: given the pair (X, G), is there an ODC of X by G?An obvious necessary condition is that X is a regular.In this paper, we are almost exclusively concerned with the starter maps of the orthogonal double covers of cayley graphs and using this method to construct ODCs by a complete bipartite graph, a complete tripartite graph, caterpillar, and a connected union of a cycle and a star whose center vertex belongs to that cycle.

Cover-Incomparability graphs (C-I graphs) form an interesting class of graphs from posets. C-I graphs are identified among chordal graphs, distance-hereditary graphs, Ptolemaic graphs, split graphs, threshold graphs, bisplit graphs, block graphs and cographs. Thus only a few classes of graphs are known to be C-I graphs so far. Composition operation and various graph products are usually used to produce more non-trivial graphs in a particular graph class, using the prime graphs in the class. In this paper, we attempt to study the effect of the composition, lexicographic and strong products of C-I graphs. We found that the lexicographic product of two C-I graphs, say G and H is a C-I graph if and only if either G is any C-I graph, and H is a complete graph or vice versa. A similar result holds for the strong product also. It can be observed that the composition operation is more general than lexicographic product and we obtain new classes of C-I graphs from this operation.

A cyber-physical system consists of two interacting networks, where the cyber network overlays the physical network. Due to the interdependence, node failures in one network may lead to failures of the other network and result in cascading failures of the whole system. One of the reasons for this phenomena is attributed to the connectivity of the interdependent system, which hinges heavily on the operation probability (or equivalently reliability) of networks in the light of node reliability. In this paper, we present the model of k- reliability, define cascading failures based on k- reliability, and propose an algorithm to calculate k -reliability for estimation of potential cascading failures. The proposed k- reliability, defined as the probability that at least k surviving nodes span an operating subnetwork, explains the operation probability of the system and reflects the connectivity of nodes in networks. In particular, we find that the network with regular interlink allocation strategy is homogeneous to the network with maximum k -reliability in class of graphs with 2 n nodes and 2 d edges when the information of intratopology is unknown. That is, regardless of the information of each individual network, the strategy that all nodes in the system are allotted the same number of bidirectional interlinks yields better performance on k- reliability than all other possible strategies both analytically and experimentally.

Fractional and circular separation dimension of graphs

For fixed graphs $F$ and $H$, a graph $G\subseteq F$ is $H$-saturated if there is no copy of $H$ in $G$, but for any edge $e\in E(F)\setminus E(G)$, there is a copy of $H$ in $G+e$. The saturation number of $H$ in $F$, denoted $sat(F,H)$, is the minimum number of edges in an $H$-saturated subgraph of $F$. In this paper, we study saturation numbers of $tK_{l,l,l}$ in complete tripartite graph $K_{n_1,n_2,n_3}$. For $t\ge 1$, $l\ge 1$ and $n_1,n_2$ and $n_3$ sufficiently large, we determine $sat(K_{n_1,n_2,n_3},tK_{l,l,l})$ exactly.

A heuristic minimal path sets method for analyzing seismic reliability of lifeline networks is proposed. For large-scale lifeline networks reliability computation can become prohibitive or inaccurate using the current enumeration methods. Although a few studies have considered both node and link failures, none of these methods has utilized the minimal paths method. This paper proposes a minimal path algorithm to evaluate reliability of relatively large lifeline networks with unreliable nodes and links. As the primary task, the method can deduce all minimal paths of a directed/undirected network. In the algorithm, minimal paths for both nodes and links are traced performing several procedures. The terminal-pair reliability of the network is then evaluated by performing an efficient Monte Carlo simulation as an alternative to exact reliability calculation methods. In terms of computational speed, the results obtained compare well with existing algorithms. The algorithm requires minimum memory storage and minimum user-defined data to represent the topology of a network. A series of case studies illustrate that the method can be used to evaluate reliability of the large-scale lifeline systems subject to node and link failures, efficiently.

Reconfiguring dominating sets in some well-covered and other classes of graphs

Accordion graphs: Hamiltonicity, matchings and isomorphism with quartic circulants

We construct biembeddings of some Latin squares which are Cayley tables of dihedral groups. These facilitate the construction of $${n^{an^2}}$$ nonisomorphic face 2-colourable triangular embeddings of the complete tripartite graph K n,n,n and the complete graph K n for linear classes of values of n and suitable constants a. Previously the best known lower bounds for the number of such embeddings that are applicable to linear classes of values of n were of the form $${2^{an^2}.}$$ We remark that trivial upper bounds are $${n^{n^2/3}}$$ in the case of complete graphs K n and $${n^{2n^2}}$$ in the case of complete tripartite graphs K n,n,n .

Bijections for Cayley trees, spanning trees, and their q-analogues

Reliability analysis of networks using stochastic model

SIZE RAMSEY NUMBERS INVOLVING MATCHINGS