I encountered the following graph, more precisely, graph structure: graph $G$ has $n$ nodes indexed from $1$ to $n$; for any $k$ between $1$ and $n$, if we remove nodes $k,k+1,\cdots,n$ and the related edges, the remaining subgraph is connected. I want to know if this is a particular class of graph structures already known and the possible application of them.
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3$\begingroup$ Trivial observation: another way to say it is that for each $1 < j \leq n$ there is at least one $i < j$ such that there is an edge $\{i,j\}$ in the graph. $\endgroup$Sam Hopkins– Sam Hopkins ♦2025-12-06 15:46:36 +00:00Commented Dec 6 at 15:46
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$\begingroup$ Isn't a graph has the property if and only if it is connected? Or are you asking about a class of ordered graphs? $\endgroup$spupyrev– spupyrev2025-12-08 23:43:31 +00:00Commented Dec 8 at 23:43
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$\begingroup$ @SamHopkins Thank you. Your comment is indeed insightful. $\endgroup$lchen– lchen2025-12-09 03:39:49 +00:00Commented Dec 9 at 3:39
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$\begingroup$ @spupyrev Yes, all connected graphs satisfy this property, but the index matters. $\endgroup$lchen– lchen2025-12-09 03:40:47 +00:00Commented Dec 9 at 3:40
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1$\begingroup$ What sort of answer do you expect? Sam Hopkins' observation shows that your "graph structures" can all be described by giving, for each $j \in \{ 2,\dots,n \}$, a non-empty subset of $\{ 1,\dots, j-1 \}$. Conversely, every such list of non-empty subsets gives one of your graph structures. $\endgroup$Tom De Medts– Tom De Medts2025-12-11 08:46:28 +00:00Commented Dec 11 at 8:46
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