Let Σ be a surface. We prove that rigidity indices of graphs which admit a polyhedral embedding in Σ and 5-connected graphs admitting an embedding in Σ are bounded by a constant depending on Σ. Moreover if the Euler characteristic of Σ is negative, then the separation index of graphs admitting a polyhedral embedding in Σ is also bounded. As a side result we show that distinguishing number of bo...

Let G be a graph. A vertex labeling of G is distinguishing if the only label-preserving automorphism of G is the identity map. The distinguishing number of G, D(G), is the minimum number of labels needed so that G has a distinguishing labeling. In this paper, we present O(n log n)-time algorithms that compute the distinguishing numbers of trees and forests. Unlike most of the previous work in t...

A coloring is distinguishing (or symmetry breaking) if no non-identity automorphism preserves it. The threshold of a graph G, denoted by θ(G), the minimum number colors k so that every k-coloring G distinguishing. We generalize this concept to edge-coloring defining an alternative index θ′(G). consider θ′ for some families graphs and find its connection with edge-cycles group. Then we show θ′(G...

In biomechanical modelling and motion analysis, the use of personalised data such as bone geometry would provide more accurate and reliable results. However, there are still a limited number of tools used to measure the evolution of articular interactions. This paper proposes a coherence index to describe the articular status of contact surfaces during motion. The index relies on a robust estim...

The paper aims at examining the properties of innovative processes with regard to Schumpeter premises on factors and drivers development economy. Due analysis changes characteristics economic entities formalised in a mathematical model, two new indexes, namely, an index creative destruction, as well innovators’ competitiveness index, are defined. As result, we get formal relationship between si...

Let c : E(G) → [k] be an edge-coloring of a graph G, not necessarily proper. For each vertex v, let c̄(v) = (a1, . . . , ak), where ai is the number of edges incident to v with color i. Reorder c̄(v) for every v in G in nonincreasing order to obtain c∗(v), the color-blind partition of v. When c∗ induces a proper vertex coloring, that is, c∗(u) 6= c∗(v) for every edge uv in G, we say that c is col...

The concept of timed-released encryption with pre-open capability (TRE-PC) was introduced by Hwang, Yum and Lee. In a TREPC scheme, a message is encrypted in such a way that it can only be decrypted at a certain point in time or if the sender releases a piece of trapdoor information known as a pre-open key. This paper examines the security model for a TRE-PC scheme, demonstrates that a TRE-PC s...

The distinguishing number D(G) of a graph G is the least cardinal number א such that G has a labeling with א labels that is only preserved by the trivial automorphism. We show that the distinguishing number of the countable random graph is two, that tree-like graphs with not more than continuum many vertices have distinguishing number two, and determine the distinguishing number of many classes...

We begin the study of distinguishing geometric graphs. Let G be a geometric graph. An automorphism of the underlying graph that preserves both crossings and noncrossings is called a geometric automorphism. A labelling, f : V (G) → {1, 2, . . . , r}, is said to be rdistinguishing if no nontrivial geometric automorphism preserves the labels. The distinguishing number of G is the minimum r such th...

The distinguishing number ∆(X) of a graph X is the least positive integer n for which there exists a function f : V (X) → {0, 1, 2, · · · , n−1} such that no nonidentity element of Aut(X) fixes (setwise) every inverse image f−1(k), k ∈ {0, 1, 2, · · · , n − 1}. All infinite, locally finite trees without pendant vertices are shown to be 2distinguishable. A proof is indicated that extends 2-disti...