Directed Convex Subgraphs

• Directed convex subgraphs are induced subdigraphs whose vertex sets remain closed under all geodesic paths, ensuring that every shortest directed path between vertices stays within the subgraph.
• They underpin key computational results by linking convexity concepts to NP-complete decision problems and informing strategies for strong orientation analysis.
• Applications in grids and modular graphs demonstrate how structured orientations yield precise convexity spectra and facilitate geometric and enumerative investigations.

Directed convex subgraphs are induced subdigraphs whose vertex sets are closed under a prescribed directed path system. In the geodesic framework for oriented graphs, which is the central graph-theoretic usage, a nonempty set $S \subseteq V(D)$ of a connected oriented graph $D$ is convex if, for every pair $x,y \in S$, the vertex set of every $xy$-geodesic and every $yx$-geodesic is contained in $S$; the induced subdigraph $D[S]$ is then a directed convex subgraph. This viewpoint leads to the convexity number $\operatorname{con}(D)$, the convexity spectrum $S_C(G)$, and the strong convexity spectrum $S_{SC}(G)$, and it supports both hardness results and exact spectrum theorems for grids (Araujo-Pardo et al., 2017). Taken together with later path-based and metric formulations, the literature suggests that directed convex subgraphs are best understood as a family of closure notions on digraphs rather than as a single universal definition.

1. Geodesic convexity in oriented graphs

Let $D$0 be an oriented graph, obtained by orienting each edge of a simple undirected graph in exactly one direction. The directed distance from $D$1 to $D$2, denoted $D$3, is the length of a shortest directed $D$4-path, and a $D$5-geodesic is any directed path of length $D$6. A nonempty set $D$7 is convex if for every pair $D$8, every vertex on every $D$9-geodesic and every $x,y \in S$0-geodesic lies in $x,y \in S$1. Equivalently,

$x,y \in S$2

The convex hull $x,y \in S$3 of a nonempty set $x,y \in S$4 is the intersection of all convex sets containing $x,y \in S$5; thus $x,y \in S$6 exactly when $x,y \in S$7 is convex (Araujo-Pardo et al., 2017).

Within this framework, a directed convex subgraph is simply the induced subdigraph $x,y \in S$8 where $x,y \in S$9 is convex. Such a subgraph has the defining property that no geodesic between vertices of $xy$0 is forced to leave $xy$1. If $xy$2 is a nontrivial connected oriented graph, the convexity number is

$xy$3

Every singleton is convex, so $xy$4. For a connected undirected graph $xy$5, the convexity spectrum and strong convexity spectrum are

$xy$6

and

$xy$7

If $xy$8 admits no strong orientation, then $xy$9. Each $yx$0 therefore records the existence of a strong orientation of $yx$1 having a proper directed convex subgraph on exactly $yx$2 vertices.

2. Structural consequences in strong orientations

Convexity in strong orientations is tightly constrained. If $yx$3 is strongly connected and $yx$4 is convex with $yx$5, then the induced subdigraph $yx$6 is strongly connected. A direct consequence is that if $yx$7 admits a strong orientation, then any proper convex set of size at least $yx$8 induces a strong subdigraph of that orientation. In particular, if $yx$9 is the undirected girth of $S$0, then

$S$1

because $S$2 must contain a directed cycle and its length is at least the length of an undirected cycle in $S$3 (Araujo-Pardo et al., 2017).

Maximal convex sets also impose a global separation structure. If $S$4 is a maximal convex set of an oriented graph $S$5, then the complement $S$6 is connected as an undirected graph. Thus a proper maximal directed convex subgraph decomposes the vertex set into two connected blocks: the convex block itself and a connected complement.

In triangle-free graphs, including grids, convexity also forces a local orientation pattern along the boundary of a convex set. If $S$7 is convex, $S$8, and $S$9 is adjacent to some $D[S]$0, then all neighbors of $D[S]$1 inside $D[S]$2 must be oriented uniformly with respect to $D[S]$3: if $D[S]$4, then $D[S]$5; if $D[S]$6, then $D[S]$7. The reason is that a mixed in/out pattern would create a two-step geodesic passing through $D[S]$8 between vertices of $D[S]$9, forcing $\operatorname{con}(D)$0 into the convex hull.

A sharp upper-extremal characterization is supplied by a result recalled from Chartrand–Fink–Zhang: for a connected oriented graph $\operatorname{con}(D)$1 of order $\operatorname{con}(D)$2,

$\operatorname{con}(D)$3

if and only if $\operatorname{con}(D)$4 contains a source, a sink, or a transitive vertex. Here a transitive vertex $\operatorname{con}(D)$5 satisfies $\operatorname{con}(D)$6, $\operatorname{con}(D)$7, and for every $\operatorname{con}(D)$8 and $\operatorname{con}(D)$9 the arc $S_C(G)$0 is present. Consequently, a strong orientation with no source, no sink, and no transitive vertex cannot realize convexity number $S_C(G)$1.

3. Complexity, hardness, and algorithmic contrasts

The decision problem Oriented Convexity Number asks, given an oriented graph $S_C(G)$2 and an integer $S_C(G)$3, whether $S_C(G)$4 contains a convex set of size at least $S_C(G)$5. This problem is in $S_C(G)$6, and it is $S_C(G)$7-complete even for bipartite oriented graphs of arbitrary large girth (Araujo-Pardo et al., 2017). The reduction is from Clique. Given a connected graph $S_C(G)$8 and $S_C(G)$9, the construction replaces each vertex $S_{SC}(G)$0 by a directed $S_{SC}(G)$1-cycle $S_{SC}(G)$2 with distinguished antipodal vertices $S_{SC}(G)$3 and $S_{SC}(G)$4, adds arcs $S_{SC}(G)$5 and $S_{SC}(G)$6 for each edge $S_{SC}(G)$7, and attaches every hexagon to a directed path

$S_{SC}(G)$8

by arcs $S_{SC}(G)$9 and $D$00. The resulting digraph is bipartite, strongly connected, and has girth $D$01, and the key identity is

$D$02

Hence $D$03 if and only if $D$04. The same idea extends to arbitrary large girth by replacing the directed $D$05-cycles with directed $D$06-cycles and the path $D$07 with a directed path of length $D$08.

This hardness result is specific to the geodesic-convexity setting on general oriented graphs. In acyclic digraphs, the literature also studies a different directed convexity: a nonempty set $D$09 is convex if there is no directed path between vertices of $D$10 which contains a vertex not in $D$11. If, additionally, the underlying undirected graph of the induced subgraph on $D$12 is connected, then $D$13 is a connected convex set, or cc-set. For this DAG setting, all connected convex sets can be enumerated in time $D$14, where $D$15 is the number of connected convex sets, and all convex sets can be enumerated in optimal time

$D$16

(0712.2661).

The contrast is substantial. In the strong-orientation geodesic model, even deciding whether a large proper directed convex subgraph exists is $D$17-complete. In the acyclic path-closure model, enumeration is output-sensitive and near-optimal or optimal. This suggests that the computational profile of directed convex subgraphs depends strongly on the path system used to define convexity.

4. Grids and the strong convexity spectrum

The strongest exact results concern grids $D$18. For the $D$19-row grid,

$D$20

Thus all even values from $D$21 up to $D$22 occur, except $D$23, and the only odd value is $D$24. For the $D$25-row grid,

$D$26

Here $D$27 is excluded by the Chartrand–Fink–Zhang theorem, while $D$28 and $D$29 are excluded because there is no connected $D$30- or $D$31-vertex subgraph of the grid that both admits a strong orientation and whose complement is connected; $D$32 is excluded by an explicit structural argument (Araujo-Pardo et al., 2017).

For general grids with $D$33 and $D$34, the strong convexity spectrum is completely determined:

$D$35

$D$36

and

$D$37

The missing values near the top end arise from general exclusions such as $D$38 and, when $D$39, also $D$40.

The central constructive devices are the whirlpool and the anti-whirlpool. In a whirlpool orientation of a rectangular subgrid, each $D$41-cycle is oriented as a directed cycle in a checkerboard pattern, and the whole region is strongly connected. For a whirlpool $D$42, one has

$D$43

because any two distinct vertices have convex hull equal to the entire region. The grid constructions then use whirlpools as atomic convexity blocks: rectangular regions whose internal geodesic structure is so tight that any proper convex set meeting such a block nontrivially must either contain the whole block or expand to the whole graph. Lemmas in the grid analysis systematically construct strong orientations with exact convexity numbers $D$44, any value $D$45 with $D$46, $D$47, $D$48, any value $D$49 for $D$50, and a wide range of values of the form $D$51.

The resulting morphology of directed convex subgraphs in grids is correspondingly rigid. Large convex sets are typically rectangular subgrids oriented as whirlpools or slight variants, and interfaces between blocks are oriented so that geodesics between vertices inside a designated block do not benefit from leaving it. This suggests a block-like theory of convexity for oriented grids: internal closure is enforced by the whirlpool geometry, while boundary arcs prevent uncontrolled convex-hull growth.

5. Short-path convexities and convex geometries on digraphs

A different branch of the subject defines directed convexity through directed paths on three vertices. In the $D$52-convexity of an oriented graph $D$53, a set $D$54 is convex if no vertex outside $D$55 is the central vertex of a directed path $D$56 with both endpoints $D$57 in $D$58. In the $D$59-convexity, the same condition is imposed only for induced directed $D$60-vertex paths, that is, paths $D$61 with $D$62. In both cases the hull is obtained by iterating the corresponding interval operator until closure, and a convexity is a convex geometry if every convex set is the hull of its extreme elements (Araújo et al., 23 Jun 2026).

The extreme vertices differ between the two models. In $D$63-convexity, a vertex is extreme if and only if it is a source or a sink. In $D$64-convexity, a vertex is extreme if and only if it is a source, a sink, or a transitive vertex. For $D$65, any vertex added during hull iteration is never extreme in the final hull.

The $D$66 case admits a clean structural characterization. An oriented graph is a convex geometry in the $D$67-convexity if and only if it is acyclic, every descendant lies at directed distance at most $D$68 from its ancestor, and whenever the graph contains an induced $D$69 with endpoints $D$70 and internal vertices $D$71, one has $D$72. This yields a polynomial-time recognition algorithm.

The $D$73 case is substantially harder. An oriented graph is a convex geometry in the $D$74-convexity if and only if it is a $D$75-free DAG and every induced obstruction from a certain family $D$76 has geometric hull in the ambient graph. Deciding whether a directed acyclic graph is a convex geometry in the $D$77-convexity is coNP-complete. On the other hand, for the hereditary class of acyclic indifference oriented graphs, geometricity in the $D$78-convexity is equivalent to being $D$79-free, and recognition can be done in $D$80 time.

These results clarify a recurrent misconception: even very local directed path rules do not lead to a unique convexity theory. Closure under all directed $D$81-paths, closure under induced directed $D$82-paths, and closure under all directed paths in DAGs produce different extreme points, different obstruction sets, and different complexity landscapes.

6. Metric, geometric, and enumerative extensions

Directed convex subgraphs also appear in more geometric and algebraic settings. In the theory of oriented modular graphs, a set is $D$83-convex if it contains every metric interval $D$84 between its vertices, and in modular graphs $D$85-convexity coincides with gatedness. For an oriented modular graph $D$86, the indicator function of a set $D$87 is L-convex if and only if $D$88 is $D$89-convex; equivalently, convex subgraphs are precisely the $D$90-valued L-convex functions. This embeds directed convex subgraphs into the broader theory of submodular functions on modular semilattices, Lovász extensions, and CAT(0) orthoscheme complexes (Hirai, 2016).

A geometric-graph analogue arises in quadrant-restricted Yao graphs. For $D$91, the subgraph defined by one quadrant is planar, is a forest, is generally disconnected, and its directed version is a directed spanner with stretch $D$92. By contrast, $D$93, the subgraph defined by two adjacent quadrants, is always connected, may be nonplanar, and is not a directed spanner. Here the “convex regions” are Euclidean quadrants and half-planes rather than graph-geodesic intervals, but the same theme reappears: enlarging the admissible directional region improves connectivity while degrading structural rigidity (0905.2249).

An enumerative extension is provided by a recent orientation–subgraph correspondence. For any graph $D$94 and sets of ordered vertex pairs $D$95, the number of $D$96-valid orientations equals the number of $D$97-valid subgraphs, where $D$98 encodes forbidden reachability and $D$99 encodes required reachability. Parallel equidistributions also hold for cycle-, cocycle-, and cycle–cocycle-reversal classes, corresponding respectively to forests, $x,y \in S$00-connected subgraphs, and $x,y \in S$01-connected forests (Bernardi et al., 15 May 2026). This suggests an enumerative route for directed-convexity notions that can be formulated as finite reachability constraints, although that step is an interpretation rather than an explicit theorem of the paper.

Across these frameworks, the common invariant is closure under directed intermediates: geodesic intermediates in strong orientations, path intermediates in DAGs, central vertices of short directed paths, metric intervals in oriented modular graphs, or cone-restricted nearest neighbors in geometric graphs. What changes from framework to framework is the ambient path system and, with it, the balance among structure, tractability, and expressive power.