a set $wsubseteq v(g)$ is called a resolving set for $g$, if for each two distinct vertices $u,vin v(g)$ there exists $win w$ such that $d(u,w)neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. the minimum cardinality of a resolving set for $g$ is called the metric dimension of $g$, and denoted by $dim(g)$. in this paper, it is proved that in a connected graph $...

We study the question of whether, for a given class of finite graphs, one can define, for each graph of the class, a linear ordering in monadic second-order logic, possibly with the help of monadic parameters. We consider two variants of monadic second-order logic: one where we can only quantify over sets of vertices and one where we can also quantify over sets of edges. For several special cas...

The relationship between graph coloring and the immersion order is considered. Vertex connectivity, edge connectivity and related issues are explored. These lead to the conjecture that, if G requires at least t colors, then G must have immersed within it Kt, the complete graph on t vertices. Evidence in support of such a proposition is presented. For each fixed value of t, there can be only a f...

the emph{harary index} $h(g)$ of a connected graph $g$ is defined as $h(g)=sum_{u,vin v(g)}frac{1}{d_g(u,v)}$ where $d_g(u,v)$ is the distance between vertices $u$ and $v$ of $g$. the steiner distance in a graph, introduced by chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. for a connected graph $g$ of order at least $2$ and $ssubseteq v(g)$, th...

In recent work [5, 6, 1] we formalize a propositional logic over partial orders. Formulæ in this logic are just like usual propositional formulæ except that propositions are statements about a partial order on a finite set of symbols. For example, (f = g)∧ ((f > h) ∨ (h > g)) is a formula in this logic. We refer to the formulæ of this logic as partial order constraints. There are many applicati...

We call T = (G1, G2, G3) a graph-triple of order t if the Gi are pairwise non-isomorphic graphs on t non-isolated vertices whose edges can be combined to form Kt. If m ≥ t, we say T divides Km if E(Km) can be partitioned into copies of the graphs in T with each Gi used at least once, and we call such a partition a T -multidecomposition. In this paper, we study multidecompositions of Km for grap...

A large class of neural network models have their units organized in a lattice with fixed topology or generate their topology during the learning process (usually unsupervised). These network models can be used as neighborhood preserving map and some of them generate a perfect topology preserving map of the input manifold using competitive Hebbian rule. But such a structure is difficult to mana...

We develop a new algorithmic technique that allows to transfer some constant time approximation algorithms for general graphs into random order streaming algorithms. We illustrate our technique by proving that in random order streams with probability at least 2/3, • the number of connected components of G can be approximated up to an additive error of εn using (1ε ) O(1/ε) space, • the weight o...

We consider the problem of characterizing small geometric graphs whose structure uniquely determines the order type of its vertex set. We describe a set of edges that prevent the order type from changing by continuous movement and identify properties of the resulting graphs.