Baker [1] devised a powerful technique to obtain approximation schemes for various problems restricted to planar graphs. Her technique can be directly extended to various other graph classes, among the most general ones the graphs avoiding a fixed apex graph as a minor. Further generalizations (e.g., to all proper minor closed graph classes) are known, but they use a combination of techniques a...

A graph G is an apex graph if it contains a vertex w such that G − w is a planar graph. It is easy to see that the genus g(G) of the apex graph G is bounded above by τ − 1, where τ is the minimum face cover of the neighbors of w, taken over all planar embeddings of G−w. The main result of this paper is the linear lower bound g(G) ≥ τ/160 (if G − w is 3-connected and τ > 1). It is also proved th...

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A graph G is an apex graph if it contains a vertex w such that G w is a planar graph It is easy to see that the genus g G of the apex graph G is bounded above by where is the minimum face cover of the neighbors of w taken over all planar embeddings of G w The main result of this paper is the linear lower bound g G if G w is connected and It is also proved that computing the genus of apex graphs...

Background and Aim: Preparation of root canal with rotary nickel-titanium (NiTi) instruments may potentially create microcracks that can lead to vertical root fracture. This study aimed to compare the incidence of crack formation by BioRaCe and Edge Taper Platinum systems in the mesiobuccal canals of extracted first mandibular molars. Materials and Methods: In this experimental study, 50 first...

Amputated apices from vegetative wildtype cells of the uninucleate green alga Acetabularia acetabulum can differentiate a reproductive structure of 'cap' in the absence of the nucleus (Hämmerling, J. (1932) Biologisches Zentralblatt 52, 42-61). To define the limits of the ability of wildtype cells to control reproductive differentiation, we determined when during development apices from wildtyp...

The mammalian cochlear duct is tonotopically organized such that the basal cochlea is tuned to high frequency sounds and the apical cochlea to low frequency sounds. In an effort to understand how this tonotopic organization is established, we searched for genes that are differentially expressed along the tonotopic axis during neonatal development. Cochlear tissues dissected from P0 and P8 mice ...

The clique cover width of G, denoted by ccw(G), is the minimum value of the bandwidth of all graphs that are obtained by contracting the cliques in a clique cover of G into a single vertex. For i = 1, 2, ..., d, let Gi be a graph with V (Gi) = V , and let G be a graph with V (G) = V and E(G) = ∩di=1(Gi), then we write G = ∩di=1Gi and call each Gi, i = 1, 2, ..., d a factor of G. The case where ...

This paper investigates response of triangular shell strip footings situated on the sandy slope. A series of reduced-scale plate load tests were conducted to cover different parameters including three shell footing types with different apex angles in addition to a flat footing, four different distances for strip footings from the crest of the slope namely “edge distance” and reinforcement statu...