5,975 research outputs found
K4-free graphs as a free algebra
Graphs of treewidth at most two are the ones excluding the clique with four vertices (K4) as a minor, or equivalently, the graphs whose biconnected components are series-parallel. We turn those graphs into a finitely presented free algebra,answering positively a question by Courcelle and Engelfriet, in the case of treewidth two. First we propose a syntax for denoting these graphs: in addition to parallel composition and series composition, it suffices to consider the neutral elements of those operations and a unary transpose operation. Then we give a finite equationa lpresentation and we prove it complete: two terms from the syntax are congruent if and only if they denote the same graph
Asymptotic Structure of Graphs with the Minimum Number of Triangles
We consider the problem of minimizing the number of triangles in a graph of
given order and size and describe the asymptotic structure of extremal graphs.
This is achieved by characterizing the set of flag algebra homomorphisms that
minimize the triangle density.Comment: 22 pages; 2 figure
- …