Can't understand K-Truss Graph properties

Cross-posted on Operations Research SE.

I'm trying to understand K-Truss Graphs which are defined as such

The k-truss is a subset of the graph with the same number of vertices, where each edge appears in at least $$𝑘 − 2$$ triangles in the original graph.

Given this example:

The $$4$$-Truss should be $$2-1-4$$ since we want edges present in at least $$2$$ triangles of the original graph:

• edge $$1-2$$ is present in triangle $$0-1-2$$ and triangle $$1-2-4.$$
• edge $$1-4$$ is present in triangle $$1-3-4$$ and triangle $$1-2-4.$$

Is it correct?

• Ummm... what is your question? Commented Aug 18, 2021 at 23:05
• I'm wondering if my solution on the given example matches the definition and its properties Commented Aug 18, 2021 at 23:08

Your solution is consistent with the definition you gave. But note that the $$k$$-truss of a graph $$G = (V,E)$$ is usually defined as the maximal subgraph $$H = (V',E')$$ such that every edge of $$H$$ is in at least $$k-2$$ triangles within $$H$$ (not in the original graph). According to this definition, the $$4$$-truss of your example graph is empty.
• Yes! The $3$-truss is just the union of the triangles in the graph; in this case, the whole graph. Commented Aug 19, 2021 at 15:12