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Traditionally needlepoint designs completely cover the canvas. [1] Although needlepoint may be worked in a variety of stitches, many needlepoint designs use only a simple tent stitch and rely upon color changes in the yarn to construct the pattern. Needlepoint is the oldest form of canvas work.
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An independent set of ⌊ ⌋ vertices (where ⌊ ⌋ is the floor function) in an n-vertex triangle-free graph is easy to find: either there is a vertex with at least ⌊ ⌋ neighbors (in which case those neighbors are an independent set) or all vertices have strictly less than ⌊ ⌋ neighbors (in which case any maximal independent set must have at least ⌊ ⌋ vertices). [4]
In the mathematical field of graph theory, the Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, chromatic number 4, and crossing number 5. It is named after German mathematician Herbert Grötzsch , who used it as an example in connection with his 1959 theorem that planar triangle-free graphs are 3-colorable.
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Many important graph families can be characterized in terms of the induced paths or cycles of the graphs in the family. Trivially, the connected graphs with no induced path of length two are the complete graphs, and the connected graphs with no induced cycle are the trees. A triangle-free graph is a graph with no induced cycle of length three.
Applying the Mycielskian repeatedly, starting with the one-edge graph, produces a sequence of graphs M i = μ(M i−1), sometimes called the Mycielski graphs. The first few graphs in this sequence are the graph M 2 = K 2 with two vertices connected by an edge, the cycle graph M 3 = C 5 , and the Grötzsch graph M 4 with 11 vertices and 20 edges.
A Markov blanket of a random variable in a random variable set = {, …,} is any subset of , conditioned on which other variables are independent with : . It means that contains at least all the information one needs to infer , where the variables in are redundant.
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