Ideally Connected Graphs

Updated 12 July 2026

Ideally connected graphs are defined so that for every pair of distinct vertices, the number of internally disjoint paths equals the minimum of their degrees.
They play a vital role in extremal connectivity problems, establishing tight bounds on average connectivity and characterizing key graph classes like cographs and chordal graphs.
Their structural analysis leverages decomposition techniques such as cotree and clique-cut decompositions, with significant implications for fault-tolerant network routing.

An ideally connected graph is a graph in which the local connectivity between every two distinct vertices is as large as the endpoint degrees permit. If $\kappa_G(u,v)$ denotes the maximum number of internally disjoint $u$ – $v$ paths in a graph $G$ , then a classical consequence of Menger’s theorem gives $\kappa_G(u,v)\le \min\{\deg_G(u),\deg_G(v)\}$ . The graph is ideally connected when equality holds for every pair of distinct vertices. In that sense, it realizes the “best-possible” local connectivity allowed by degree constraints. The notion also appears under the names maximally locally connected and strongly Menger–connected, and it has been studied both structurally and in connection with extremal average-connectivity problems and routing in fault-tolerant networks (Jordaan, 17 Sep 2025).

1. Definition and immediate consequences

Let $G=(V,E)$ be a simple graph and let $u\ne v$ be vertices of $G$ . The parameter

$\kappa_G(u,v)$

is the maximum number of internally disjoint $u$ – $u$ 0 paths in $u$ 1. In the broader multigraph setting one also considers

$u$ 2

the maximum number of edge-disjoint $u$ 3– $u$ 4 paths. The standard inequality chain is

$u$ 5

for all distinct $u$ 6 (Casablanca et al., 2018).

A graph is ideally connected if

$u$ 7

Analogously, a graph is ideally edge-connected if

$u$ 8

An ideally connected graph is therefore automatically ideally edge-connected (Casablanca et al., 2018).

The condition is local in formulation but global in force. It requires every pair of vertices to admit a saturating family of internally disjoint paths, so the graph cannot have even a single pair for which the degree bound is not attained. This is the feature that makes the class rigid enough to admit exact characterizations inside several hereditary graph classes (Jordaan, 17 Sep 2025).

2. Extremal role in average connectivity

Ideal connectedness is closely tied to average-connectivity extremal problems. For a graph $u$ 9 of order $v$ 0, the average connectivity is

$v$ 1

and the average edge-connectivity is

$v$ 2

To compare these quantities with the degree sequence, one defines the potential

$v$ 3

If $v$ 4 has degree sequence $v$ 5, then

$v$ 6

with equality if and only if $v$ 7 is ideally connected (Casablanca et al., 2018).

This converts ideal connectedness into an extremal equality case for a natural degree-based upper bound. The same work shows that among all nonnegative integer sequences with fixed total sum, the unique maximizer of the potential is a nearly regular sequence. Moreover, for every integer pair

$v$ 8

there exists a simple nearly regular graph of order $v$ 9, size $G$ 0, and which is ideally connected (Casablanca et al., 2018).

These facts underpin the structure of extremal minimally $G$ 1-connected graphs. Every minimally $G$ 2-connected graph of order $G$ 3 with largest average connectivity is bipartite, with the set of vertices of degree $G$ 4 and the set of vertices of degree at least $G$ 5 being the partite sets. The universal bound

$G$ 6

holds for every minimally $G$ 7-connected graph, it is asymptotically tight, and every extremal graph of order $G$ 8 is obtained from some ideally connected nearly regular graph on roughly $G$ 9 vertices and $\kappa_G(u,v)\le \min\{\deg_G(u),\deg_G(v)\}$ 0 edges by subdividing every edge (Casablanca et al., 2018). An analogous statement holds for average edge-connectivity in minimally $\kappa_G(u,v)\le \min\{\deg_G(u),\deg_G(v)\}$ 1-edge-connected graphs.

3. Cographs

A cograph is a $\kappa_G(u,v)\le \min\{\deg_G(u),\deg_G(v)\}$ 2-free graph; equivalently, it is the smallest class containing the single-vertex graph and closed under disjoint union and complete join. Within this class, ideal connectedness admits an exact forbidden-subgraph characterization: a cograph $\kappa_G(u,v)\le \min\{\deg_G(u),\deg_G(v)\}$ 3 is ideally connected if and only if $\kappa_G(u,v)\le \min\{\deg_G(u),\deg_G(v)\}$ 4 is $\kappa_G(u,v)\le \min\{\deg_G(u),\deg_G(v)\}$ 5-free (Jordaan, 17 Sep 2025).

The proof proceeds by induction on the unique cotree decomposition of the cograph. In the join-case, if $\kappa_G(u,v)\le \min\{\deg_G(u),\deg_G(v)\}$ 6, then $\kappa_G(u,v)\le \min\{\deg_G(u),\deg_G(v)\}$ 7 and $\kappa_G(u,v)\le \min\{\deg_G(u),\deg_G(v)\}$ 8 are ideally connected if and only if $\kappa_G(u,v)\le \min\{\deg_G(u),\deg_G(v)\}$ 9 is ideally connected. The forward direction uses the paths already present in each side together with length- $G=(V,E)$ 0 detours through the opposite side; the reverse direction shows that ideal connectedness of the join restricts back to ideal connectedness of each factor. In the union-case, if $G=(V,E)$ 1 and $G=(V,E)$ 2 is ideally connected, then one of $G=(V,E)$ 3 must be edgeless; otherwise vertices chosen from different components would satisfy $G=(V,E)$ 4 while both have positive degree (Jordaan, 17 Sep 2025).

The obstruction $G=(V,E)$ 5 is minimal and transparent. If $G=(V,E)$ 6 and $G=(V,E)$ 7 lie on different edges of an induced $G=(V,E)$ 8, then $G=(V,E)$ 9 but $u\ne v$ 0, so the degree bound fails. Conversely, the decomposition analysis shows that forbidding induced $u\ne v$ 1 is sufficient inside the cograph class (Jordaan, 17 Sep 2025).

4. Chordal graphs and threshold structure

A graph is chordal if every induced cycle of length at least $u\ne v$ 2 has a chord. In this class, the ideally connected graphs are precisely the threshold graphs (Jordaan, 17 Sep 2025). A threshold graph is one built from $u\ne v$ 3 by repeatedly adding either an isolated vertex or a dominating vertex. Equivalently, it is exactly the class of $u\ne v$ 4-free graphs, or equivalently split graphs whose independent side has nested neighborhoods (Jordaan, 17 Sep 2025).

The two main exact classifications can be summarized as follows.

Graph class	Ideally connected members	Equivalent description
Cographs	Exactly the $u\ne v$ 5-free cographs	No induced $u\ne v$ 6 (Jordaan, 17 Sep 2025)
Chordal graphs	Exactly the threshold graphs	$u\ne v$ 7-free split graphs with nested neighborhoods (Jordaan, 17 Sep 2025)

The chordal proof uses clique-cut decompositions. Dirac’s theorem implies that in a noncomplete chordal graph every minimum vertex-cutset is a clique. Writing such a cutset as $u\ne v$ 8 with $u\ne v$ 9, the decomposition of $G$ 0 along $G$ 1 into $G$ 2-attached chordal subgraphs is constrained sharply by ideal connectedness: exactly one $G$ 3-subgraph has connectivity at least $G$ 4 and is itself ideally connected and chordal, while every other $G$ 5-subgraph consists of $G$ 6 plus one extra vertex $G$ 7 with $G$ 8 (Jordaan, 17 Sep 2025).

This forces a split decomposition with nested neighborhoods, hence threshold structure. The converse is immediate from the cograph result, since every threshold graph is a $G$ 9-free cograph and therefore ideally connected (Jordaan, 17 Sep 2025).

5. Examples, non-examples, and local path geometry

Several standard graph families illustrate the definition sharply. Any complete graph $\kappa_G(u,v)$ 0 is trivially ideally connected. Any star $\kappa_G(u,v)$ 1 is ideally connected. At the opposite extreme, $\kappa_G(u,v)$ 2 is a cograph but not ideally connected, and a split graph whose independent vertices have non-nested neighborhoods is chordal but not ideally connected (Jordaan, 17 Sep 2025).

The contrast between threshold and non-threshold chordal graphs is especially instructive. A threshold graph admits a recursive construction by isolated and dominating vertices, and this recursive simplicity is reflected in its path structure. In fact, threshold graphs are not only ideally connected: between any two vertices $\kappa_G(u,v)$ 3 the required $\kappa_G(u,v)$ 4 internally disjoint paths can all be found of length at most $\kappa_G(u,v)$ 5 (Jordaan, 17 Sep 2025).

This short-path property gives a concrete interpretation of the equality $\kappa_G(u,v)$ 6. In threshold graphs, saturation of the smaller endpoint degree does not require long detours or globally complicated routing; the graph’s nested neighborhood structure realizes the disjoint path system in a highly compressed form. By contrast, the presence of non-nested neighborhoods in a split graph creates exactly the kind of interference that prevents simultaneous saturation for all vertex pairs (Jordaan, 17 Sep 2025).

The structural theory of ideally connected graphs in cographs and chordal graphs is driven by decomposition. For cographs, the central tools are the union and join operations together with induction on the cotree. For chordal graphs, the crucial ingredient is that every minimal separator is a clique, so clique-cut analysis reduces ideal connectedness to a single “big” piece plus simplicial attachments. In both settings, Menger’s theorem and saturation arguments control the equality cases in

$\kappa_G(u,v)$ 7

and force the eventual forbidden-subgraph descriptions (Jordaan, 17 Sep 2025).

A recurrent source of confusion is terminological rather than mathematical. Ideally connected graphs concern local path connectivity. By contrast, distance ideals are algebraic invariants that generalize the Smith normal form and the spectrum of several distance matrices associated with a graph (Alfaro et al., 16 Apr 2025). Likewise, the $\kappa_G(u,v)$ 8-connected ideal of a graph is a square-free monomial ideal generated by the connected induced subgraphs with $\kappa_G(u,v)$ 9 vertices, and when $u$ 0 it coincides with the usual edge ideal (Chau et al., 5 Dec 2025). A related formulation studies an $u$ 1-connected ideal generated by connected induced subgraphs of size $u$ 2 (Ghosh et al., 6 Dec 2025). These are separate theories despite the overlap in vocabulary.

The current structural classifications also point to broader directions. Suggested extensions include larger super-classes of cographs, such as distance-hereditary graphs, parity graphs, or circle graphs, asking which members are ideally connected; further variants include strongly $u$ 3-Menger-connected versions and edge-versions of the problem (Jordaan, 17 Sep 2025). In this sense, the known cograph and chordal results serve both as exact classification theorems and as templates for a wider decomposition-based program in local connectivity theory.