A note on the unit distance problem for planar configurations with $\mathbb{Q}$-independent direction set

The Maximum Independent Set Problem in Planar Graphs

We study the computational complexity of finding a maximum independent set of vertices in a planar graph. In general, this problem is known to be NP-hard. However, under certain restrictions it becomes polynomial-time solvable. We identify a graph parameter to which the complexity of the problem is sensible and produce a number of both negative (intractable) and positive (solvable in polynomial...

A note on greedy algorithms for the maximum weighted independent set problem

In this paper, we consider three simple and natural greedy algorithms for the Maximum Weighted Independent Set problem. We show that two of them output an independent set of weight P v2V (G) W (v) d(v)+1 and the other outputs an independent set of weight P v2V (G) W (v) 2 P u2N + G (v) W (u) . Those results are generalization of Tur an's theorem.

the algorithm for solving the inverse numerical range problem

برد عددی ماتریس مربعی a را با w(a) نشان داده و به این صورت تعریف می کنیم w(a)={x8ax:x ?s1} ، که در آن s1 گوی واحد است. در سال 2009، راسل کاردن مساله برد عددی معکوس را به این صورت مطرح کرده است : برای نقطه z?w(a)، بردار x?s1 را به گونه ای می یابیم که z=x*ax، در این پایان نامه ، الگوریتمی برای حل مساله برد عددی معکوس ارانه می دهیم.

The Unit Distance Problem on Spheres

Note on the Homogeneous Set Sandwich Problem

A homogeneous set is a non-trivial module of a graph, i.e., a non-unitary, proper subset H of a graph’s vertices such that all vertices in H have the same neighbors outside H . Given two graphs G1(V ,E1), G2(V ,E2), the Homogeneous Set Sandwich Problem asks whether there exists a sandwich graph GS(V,ES), E1 ⊆ES ⊆E2, which has a homogeneous set. Recently, Tang et al. [Inform. Process. Lett. 77 (...