A graph G is list point k-arborable if, whenever we are given a k-list assignment L(v) of colors for each vertex v ∈ V(G), we can choose a color c(v) ∈ L(v) for each vertex v so that each color class induces an acyclic subgraph of G, and is equitable list point k-arborable if G is list point k-arborable and each color appears on at most ⌈|V(G)|/k⌉ vertices of G. In this paper, we conjecture tha...

In this paper, we study graphs with integer matching polynomial roots. We characterize all traceable graphs whose all matching polynomial roots are integer. We show that there is no k-regular claw-free graph (k ≥ 2), whose all matching polynomial roots are integer. Also we conjecture that apart from K7 \ (E(C3) ∪E(C4)), there is no matching integral k-regular graph (k ≥ 2). 2010 Mathematics Sub...

Given a graph G, its partially square graph G∗ is a graph obtained by adding an edge (u, v) for each pair u, v of vertices of G at distance 2 whenever the vertices u and v have a common neighbor x satisfying the condition NG(x) ⊆ NG[u] ∪ NG[v], where NG[x] = NG(x) ∪ {x}. In the case where G is a claw-free graph, G∗ is equal to G. We define σ◦ t = min{ ∑ x∈S dG(x) : S is an independent set in G ...

We study the class of independence complexes of claw-free graphs. The main theorem give good bounds on the connectivity of these complexes, given bounds for a few subcomplexes of the same class. Two applications are presented. Firstly, we show that the independence complex of a claw-free graph with n vertices and maximal degree d is (cn/d + ε)–connected, where c = 2/3. This can be compared with...

Seymour conjectured for a fixed integer k ≥ 2 that if G is a graph of order n with δ(G) ≥ kn/(k + 1), then G contains the kth power Ck n of a Hamiltonian cycle Cn of G, and this minimum degree condition is sharp. Earlier the k = 2 case was conjectured by Pósa. This was verified by Komlós et al. [4]. For s ≥ 3, a graph is K1,s-free if it does not contain an induced subgraph isomorphic to K1,s. S...

In this paper we show how to solve the Maximum Weight Stable Set Problem in a claw-free graph G(V,E) with α(G) ≤ 3 in time O(|E| log |V |). More precisely, in time O(|E|) we check whether α(G) ≤ 3 or produce a stable set with cardinality at least 4; moreover, if α(G) ≤ 3 we produce in time O(|E| log |V |) a maximum stable set of G. This improves the bound of O(|E||V |) due to Faenza et alii ([2]).

Finding augmenting chains is in the heart of the maximum matching problem, which is equivalent to the maximum stable set problem in the class of line graphs. Due to the celebrated result of Edmonds, augmenting chains can be found in line graphs in polynomial time. Minty and Sbihi generalized this result to claw-free graphs. In this paper we extend it to larger classes. As a particular consequen...

A set S of vertices in graph [Formula: see text] is a [Formula: see text], abbreviated TRDS, of G if every vertex of G is adjacent to a vertex in S and every vertex of [Formula: see text] is adjacent to a vertex in [Formula: see text]. The [Formula: see text] of G, denoted by [Formula: see text], is the minimum cardinality of a TRDS of G. Jiang and Kang (J Comb Optim. 19:60-68, 2010) characteri...

Normal hand function is a balance between the extrinsic-intrinsic and extensor-flexor group of musculature. Although individually the intrinsics are very small muscles, collectively they contribute about 50% of grip strength. Total 19 patents with claw deformity were corrected by 4 different techniques. 11 claws were due to high ulnar nerve palsy and 8 were due to low palsy. Result was excellen...

CLAW is a handheld virtual reality controller that augments the typical controller functionality with force feedback and actuated movement to the index finger. Our controller enables three distinct interactions (grasping virtual object, touching virtual surfaces, and triggering) and changes its corresponding haptic rendering by sensing the differences in the user’s grasp. A servo motor coupled ...