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It seems obvious and easy to define the tree-ness $\tau$ of an undirected graph e.g. by the minimal number $k$ of edges one must change to make the graph a tree

$$\tau = 1 - \frac{k}{N}$$

The number $N$ has to be choosen such that the complete graph - which is maximally un-tree-ish - has $\tau = 0$. That means that $N = \frac{1}{2}(n-1)(n-2)$ for a graph with $n$ nodes. This $N$ is exactly the number of edges one has to remove to make the complete $n$-graph a tree.

The same kind of definition can be applied to any other dichotomous graph property $X$ thus measuring the $X$-ness of a graph.

My question is: Is such kind of $X$-ness actually studied and plays a role, or is it a unfruitful concept?

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    Don't use the phrase "$P$-ness" out loud – 2017-01-18
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    I changed it to avoid trouble. – 2017-01-18
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    I believe this is called resilience of a graph. This paper https://www.math.ias.edu/csdm/files/05-06/bsudakov_local_resilience_of_graphs.pdf is a nice beginning. It is not widely used as far as i know. But there might be very usefull applications – 2017-06-16
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    I am definitely not an expert; that is why I write this in the comments , since I hope more of an expert can answer – 2017-06-16
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    Out of curiosity, if you have to remove one edge and add another, does that make make $k=1$ or $k=2$? The concept may be relevant in consensus theory where the necessary and sufficient convergence condition in linear consensus on static graphs is the existence of a spanning tree subgraph. But that relevance is just speculation. See Olfati-Saber's Consensus Problems in Networks of Agents With Switching Topology and Time-Delays (2004). Can't say I've seen anything like it (tree-ness) before though :/ – 2017-06-21
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    I am not sure what the word "change" means in your definition (it cannot possibly mean: remove an edge $e$ and insert another edge between the given vertex set elsewhere). The meaningful interpretation would be "remove". Then the minimal number of edges of the given connected graph needed to be removed so that the resulting graph is a tree, is usually called "the 1st Betti number of the graph", i.e. the rank of the 1st homology group of its underlying space. This notion is well-studied for the last 150 or so years. If you want to allow nonconnected graphs, replace "tree" with "forest". – 2017-06-21

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