We connect two seemingly unrelated problems in graph theory. Any graph G has a neighborhood multiset N (G) = {N(x) | x ∈ V (G)} whose elements are precisely the open vertex-neighborhoods of G. In general there exist non-isomorphic graphs G and H for which N (G) = N (H). The neighborhood reconstruction problem asks the conditions under which G is uniquely reconstructible from its neighborhood mu...

An exponential dominating set of graph $G = (V,E )$ is a subset $Ssubseteq V(G)$ such that $sum_{uin S}(1/2)^{overline{d}{(u,v)-1}}geq 1$ for every vertex $v$ in $V(G)-S$, where $overline{d}(u,v)$ is the distance between vertices $u in S$ and $v  in V(G)-S$ in the graph $G -(S-{u})$. The exponential domination number, $gamma_{e}(G)$, is the smallest cardinality of an exponential dominating set....

We consider the constrained longest common subsequence problem with an arbitrary set of input strings as well pattern strings. This has applications, for example, in computational biology where it serves a measure similarity sets molecules putative structures common. contribute several ways. First, is formally proven that finding feasible solution length is, general, NP-complete. Second, we pro...

The AutoGraphiX 2 system is used to compare the index of a connected graph G with a number of other graph theoretical invariants, i.e., chromatic number, maximum, minimum and average degree, diameter, radius, average distance, independence and domination numbers. In each case, best possible lower and upper bounds, in terms of the order of G, are sought for sums, differences, ratios and products...

Given an undirected weighted graph G = (V,E) with vertex set V, edge set E and weights wi ∈ R associated to V or to E. Minimum weighted k-Cardinality tree problem (k-CARD for short) consists of finding a subtree of G with exactly k edges whose sum of weights is minimum [4]. There are two versions of this problem: vertex-weighted and edge-weighted, if weights to V or to E are associated, respect...

‎Let $G$ be a non-abelian group and let $Gamma(G)$ be the non-commuting graph of $G$‎. ‎In this paper we define an equivalence relation $sim$ on the set of $V(Gamma(G))=Gsetminus Z(G)$ by taking $xsim y$ if and only if $N(x)=N(y)$‎, ‎where $ N(x)={uin G | x textrm{ and } u textrm{ are adjacent in }Gamma(G)}$ is the open neighborhood of $x$ in $Gamma(G)$‎. ‎We introduce a new graph determined ...

Let G be a Hamiltonian graph. A factor F of G is called a Hamiltonian factor if F contains a Hamiltonian cycle. In this paper, two sufficient conditions are given, which are two neighborhood conditions for a Hamiltonian graph G to have a Hamiltonian factor. Keywords—graph, neighborhood, factor, Hamiltonian factor.

For a simple graph G let NG(u) be the (open) neighborhood of vertex u ∈ V (G). Then G is neighborhood anti-Sperner (NAS) if for every u there is a v ∈ V (G)\{u} with NG(u) ⊆ NG(v). And a graph H is neighborhood distinct (ND) if every neighborhood is distinct, i.e., if NH(u) 6= NH(v) when u 6= v, for all u and v ∈ V (H). In Porter and Yucas [3] a characterization of regular NAS graphs was given:...