Let G be a graph. The Steiner distance of $$W\subseteq V(G)$$ is the minimum size connected subgraph containing W. Such necessarily tree called W-tree. set $$A\subseteq k-Steiner general position if $$V(T_B)\cap A = B$$ holds for every $$B\subseteq A$$ cardinality k, and B-tree $$T_B$$ . number $$\mathrm{sgp}_k(G)$$ largest in G. cliques are introduced used to bound from below. determined trees...

Abstract We study planar polygonal curves with the variational methods. show a unified interpretation of discrete curvatures and Steiner-type formula by extracting notion curvature vector from first variation length functional. Moreover, we determine equilibrium for functional under area-constraint condition their stability.

We present optimal online algorithms for two related known problems involving Steiner Arborescence, improving both the lower and the upper bounds. One of them is the well studied continuous problem of the Rectilinear Steiner Arborescence (RSA). We improve the lower bound and the upper bound on the competitive ratio for RSA from O(logN) and Ω( √ logN) to Θ( logN log logN ), where N is the number...

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...

We give a new proof of the isoperimetric inequality in plane, based on Steiner's formula for area convex neighborhood. This establishes directly, without requiring that we separately establish existence an optimal domain. In doing so, this bypasses main difficulty all proofs Steiner outlined plane inequality.

Abstract We investigate the weighted $L_p$ affine surface areas which appear in recently established Steiner formula of Brunn–Minkowski theory. show that they are valuations on set convex bodies and prove isoperimetric inequalities for them. related to f divergences cone measures body its polar, namely Kullback–Leibler divergence Rényi divergence.

The minimization problem for Horn formulas is to find a Horn formula equivalent to a given Horn formula, using a minimum number of clauses. A 2 1−ǫ(n)-inapproximability result is proven, which is the first inapproximability result for this problem. We also consider several other versions of Horn minimization. The more general version which allows for the introduction of new variables is known t...

The minimization problem for Horn formulas is to find a Horn formula equivalent to a given Horn formula, using a minimum number of clauses. A 2 1−ǫ(n)-inapproximability result is proven, which is the first non-trivial inapproximability result for this problem. We also consider several other versions of Horn minimization. The more general version which allows for the introduction of new variable...