Saturated Subgraphs of the Hypercube

Given graphs $G$ and $H$, $G$ is $H$-saturated if $H$ is not a subgraph of $G$, but for all $e \notin E(G)$, $H$ appears as a subgraph of $G + e$. While for every $n \ge |V(H)|$, there exists an $n$-vertex graph that is $H$-saturated, the same does not hold for induced subgraphs. That is, there exist graphs $H$ and values of $n \ge |V(H)|$, for which every $n$-vertex graph $G$ either contains $H$ as an induced subgraph, or there exists $e \notin E(G)$ such that $G + e$ does not contain $H$ as an induced subgraph. To circumvent this Martin and Smith make use of a generalized notion of "graph" when introducing the concept of induced saturation and the induced saturation number of graphs. This allows for edges that can be included or excluded when searching for an induced copy of $H$, and the induced saturation number is the minimum number of such edges that are required.In this paper, we show that the induced saturation number of many common graphs is zero. This yields graphs that are $H$-induced-saturated. That is, graphs such that no induced copy of $H$ exists, but adding or deleting any edge creates an induced copy of $H$. We introduce a new parameter for such graphs, indsat*($n;H$), which is the minimum number of edges in an $H$-induced-saturated graph. We provide bounds on indsat*($n;H$) for many graphs. In particular, we determine indsat*($n;H$) completely when $H$ is the paw graph $K_{1,3}+e$, and we determine indsat*(n;$K_{1,3}$) within an additive constant of four.

Three new theorems are proved in the paper, which give bounds for the number of edges in induced subgraphs of a special distance graph.

We derive a number of upper and lower bounds for the first nontrivial eigenvalue of Laplacians on combinatorial and quantum graph in terms of the edge connectivity, i.e. the minimal number of edges which need to be removed to make the graph disconnected. On combinatorial graphs, one of the bounds corresponds to a well-known inequality of Fiedler, of which we give a new variational proof. On quantum graphs, the corresponding bound generalizes a recent result of Band and Lévy. All proofs are general enough to yield corresponding estimates for the p-Laplacian and allow us to identify the minimizers.Based on the Betti number of the graph, we also derive upper and lower bounds on all eigenvalues which are ‘asymptotically correct’, i.e. agree with the Weyl asymptotics for the eigenvalues of the quantum graph. In particular, the lower bounds improve the bounds of Friedlander on any given graph for all but finitely many eigenvalues, while the upper bounds improve recent results of Ariturk. Our estimates are also used to derive bounds on the eigenvalues of the normalized Laplacian matrix that improve known bounds of spectral graph theory.

This paper introduces the notions of chained and semi-chained graphs. The chain of a graph, when existent, refines the notion of bipartivity and conveys important structural information. Also the notion of a center vertex v_c is introduced. It is a vertex, whose sum of p powers of distances to all other vertices in the graph is minimal, where the distance between a pair of vertices {v_c,v} is measured by the minimal number of edges that have to be traversed to go from v_c to v. This concept extends the definition of closeness centrality. Applications in which the center node is important include information transmission and city planning. Algorithms for the identification of approximate central nodes are provided and computed examples are presented.

On color critical graphs

A cubic graph is nonplanar if and only if it contains a subgraph homeomorphic to K 3,3 . For a given 2-connected cubic graph G , denoted by ed(G) the minimal number of edges so that after removal them from G the resulting graph becomes planar and g(G) the genus of G . Moreover, for a given simple graph G let cr(G) denote the minimal number of crossings of edges needed to draw G on the plain (so the minimum is taken over all submersions of G in the plane). In this paper, we study relations between the characteristics ed(G) and g(G) and cr(G) for some special classes of graphs and discuss the problems related with their evaluation.

For a graph G, we define c(G) to be the minimal number of edges we must delete in order to make G into a covering graph of some poset. We prove that, if p=n -1+η(n) ,where η(n) is bounded away from 0, then there is a constant k0>0 such that, for a.e. G p , c(G p )≥k0n 1+η(n) .In other words, to make G p into a covering graph, we must almost surely delete a positive constant proportion of the edges. On the other hand, if p=n -1+η(n) , where η(n)→0, thenc(G p )=o(n 1+η(n) ), almost surely.

The restricted hypercube-like graphs, variants of the hypercube, were proposed as desired interconnection networks of parallel systems. The matching preclusion number of a graph is the minimum number of edges whose deletion results in the graph with neither perfect matchings nor almost perfect matchings. The fractional perfect matching preclusion and fractional strong perfect matching preclusion are generalizations of the matching preclusion. In this paper, we obtain fractional matching preclusion number and fractional strong matching preclusion number of restricted hypercube-like graphs, which extend some known results.

Channel assignment for interference avoidance in honeycomb wireless networks

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number $${{\rm sd}_{\gamma_t}(G)}$$ is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper, we prove that $${{\rm sd}_{\gamma_t}(G)\leq 2\gamma_t(G)-1}$$for every simple connected graph G of order n ≥ 3.

A connected, nontrivial, simple graph of order v is said to be α,β destructible if α,β are factors of v and an α-set of edges, E', exists whose removal from G isolates exactly the vertices in α,β-set V'. Graphs which are not α,β destructible for any α,β are called stable, If G is a stable graph on a prime number p ≥ 7 of vertices, then we show that G has a maximum number of edges if and only if G is K2,p-2, We also characterize stable graphs on a minimum number of edges.

We study two variants of the Balanced Tree Partition (BTP for short) problem, whose goal is to find the minimum number of edges such that a partition of vertices into sets of equal size can be obtained after deleting those edges. We consider the BTP problem over trees with virtual nodes, which is motivated by the real applications in storing tree structured data distributively. Different from the traditional BTP problem, after deleting an edge from the tree, two virtual nodes must be added to provide the ability to recover the edge when computing, which will increase the size of some set in the partition and the set may become too large and need further splitting. Depending on whether or not to consider the partition set number k to be a parameter, to investigate the balanced tree partition methods over trees with virtual nodes, we formally defined two specific problems, \(k\)-VBTP and VBTP. The computational complexity and inapproximability of \(k\)-VBTP are analyzed, for the VBTP problem, a polynomial time algorithm is designed to find the optimal solution based on dynamic programming. Finally, the connections between \(k\)-VBTP and VBTP are built, and based the algorithm for VBTP, we design a heuristic algorithm for \(k\)-VBTP with performance guarantee.

A set $D$ of vertices of graph $G$ is called $double$ $dominating$ $set$ if for any vertex $v$, $|N[v]\cap D|\geq 2$. The minimum cardinality of $double$ $domination$ of $G$ is denoted by $\gamma_d(G)$. The minimum number of edges $E'$ such that $\gamma_d(G\setminus E)>\gamma_d(G)$ is called the double bondage number of $G$ and is denoted by $b_d(G)$. This paper determines that $b_d(G\vee H)$ and exact values of $b(P_n\times P_2)$, and generalized corona product of graphs.

Let $G$ be a simple finite undirected and an isolate-free graph. A set $S$ of vertices in $G$ is a semitotal dominating set of $G$ if it is a dominating set of $G$ and every vertex in $S$ is within distance 2 of another vertex of $S$. The semitotal domination number, $\gamma_{t2}(G)$, is the minimum cardinality of such a set. In this paper, we study the semitotal domination number for Harary graphs, which was first introduced by Frank Harary. Since Harary graphs have the maximum possible connectivity with the minimum number of edges, many researchers are interested in studying its stability properties.

Optimal 1-edge fault-tolerant designs for ladders