# Krull dimension of binomial edge ideal of trees

In the article C. Mascia, G. Rinaldo, Krull dimension and regularity of binomial edge ideals of block graphs the authors described an algorithm to compute the Krull dimension of binomial edge ideals of block graphs.

In this page you can download our algorithm (KrullDimTree and RenameGraph) implemented in CoCoA in the case the block graph is a tree.

Moreover we give an example of this computation step by step in the following slides: Krull dimension of a tree of the following tree

Here is an example of computation. We show the input and the output in the console.

Source "RenameGraph.coc"; Source "KrullDimTree.coc"; G:=[[1,2],[2,3],[2,10],[3,4],[3,5],[3,6],[6,7],[6,8],[6,9],[10,11],[10,12],[10,16],[12,13],[13,14],[13,15]]; Nv:=16; DimTree(RenameGrapgh(G,Nv),1,Nv);The output is enriched showing in each call of the recursive function: 1) The analyzed vertex v; 2) For each children of v if it is or not in the cutset T.

Analizing the vertex 1 The children of 1 are [2] Analizing the vertex 2 The children of 2 are [3, 10] Analizing the vertex 3 The children of 3 are [4, 5, 6] Analyzing the vertex 4 4 is a leaf and then is not in T Analyzing the vertex 5 5 is a leaf and then is not in T Analyzing the vertex 6 The children of 6 are [7, 8, 9] Analyzing the vertex 7 7 is a leaf and then is not in T Analyzing the vertex 8 8 is a leaf and then is not in T Analyzing the vertex 9 9 is a leaf and then is not in T The degree of 6 is 4 and then 6 is in T The degree of 3 is 3 and then 3 is in T Analyzing the vertex 10 The children of 10 are [11, 12, 16] Analyzing the vertex 11 11 is a leaf and then is not in T Analyzing the vertex 12 The children of 12 are [13] Analyzing the vertex 13 The children of 13 are [14, 15] Analyzing the vertex 14 14 is a leaf and then is not in T Analyzing the vertex 15 15 is a leaf and then is not in T The degree of 13 is 3 and then 13 is in T The degree of 12 is 1 and then 12 is not in T Analyzing the vertex 16 16 is a leaf and then is not in T The degree of 10 is 4 and then 10 is in T The degree of 2 is 1 and then 2 is not in T The degree of 1 is 1 and then 1 is not in T Dimension=23