Given a continuous scalar function f : ℳ → ℝ on a topological space ℳ, the Reeb graph is defined as the quotient space of ℳ induced by the equivalence relation “ ∼ ” with p ∼ q ↔ p and q belong to the same level set component (called a contour or an isosurface) [1]. If ℳ is simply connected, the Reeb graph is also simply connected and is called contourtree [10]. In the discrete settings, we consider a simplicial mesh ℳ and a function f defined by the values at the vertices and extended by linear interpolation over ℳ. We assume that f values are unique at the vertices, perhaps by perturbation of the input data, so that f is a Morse function and the critical points are isolated and occur at vertices of the mesh [11, 12]. The discrete Reeb graph consists of contours passing through the critical vertices, recording the creation, destruction, splitting, and merging of contours. It can be augmented so that the contours passing through the vertices are the nodes, which are connected by the classes of contours not passing through any vertices, called the arcs [10, 13].
In this paper, we let ℑ = < Ω, 𝔈 > be a fixed connected simple graph, on which we define the isograph and all related concepts. Unless otherwise noted, a property of an item t is denoted by propertyName(t), such as “lowValue(S)” in Section 3, with the first letter of the last word capitalized. Except the following term aliases, we adopt common terminologies of graph theory in this paper.

Each vertex in Ω is called a site.

Each edge in 𝔈 ⊆ Ω × Ω is called a surfel (short for surface element) and any subset of 𝔈 is called a surface [14]. In our formulation, surfels are oriented: (p, q) and (q, p) are inverse to each other, i.e., (p, q)− 1 = (q, p). The inverse of a surface S is S−1 = {e−1  e∈S}. To handle the orientation of surfels, we let ℑ be directed, and 𝔈 symmetric, i.e., ∀ e∈𝔈, e−1∈𝔈.
We impose no geometrical restriction on the input graph. The following geometrical interpretation is introduced to help understanding, and itself is not a requirement of the theory. A site is perceived as a voxel (short for volume element). A surfel (p, q) is interpreted as a common edge/face shared by the voxels of p and q. In this paper when depicting an input graph in a figure, we use this interpretation. See Fig. 1.
For a surface S, the immediate interior (resp. exterior), noted by II(S) (resp. IE(S)), is defined as II(S) = {p  (p, q) ∈S} (resp. IE(S) = {q  (p, q) ∈S}). A surface S is called orientable if II(S) and IE(S) are disjoint. The boundary of a subset X of Ω, noted by ∂X, is a surface defined as ∂X = {(p, q) ∈ 𝔈  p ∈ X and q ∉ X}. ∂X is always orientable.
Two surfels (p1, q1) and (p2, q2) are called orientablymeshed (or simply meshed) if (1) p1 = p2 and (q1, q2) ∈𝔈, or (2) q1 = q2 and (p1, p2) ∈𝔈 (see Fig. 2a). A surface S is called meshed if any two surfels β1 and β2 in S are connected by a meshed surfel path \({\left\{{\alpha }_{k}\right\}}_{k=1}^{m}\) ⊆ S, i.e., β1 = α1, β2 = αm, and for 1 ≤ k < m, αk and αk+1 are meshed. A maximal meshed orientable surface is called a sect of ℑ. See Fig. 2b.
For surfaces S and T, they are called tangent if S ∩ T ≠ ∅; called inversely tangent if S ∩ T−1 ≠ ∅. We say S is locally before T (or T is locally after S), written as S ≺ T (or T ≻ S), if IE(S) ∩ II(T) ≠ ∅. See Fig. 2c. Two surfaces S and T are called crossing each other if S ≺ T and T ≺ S. Sects are not selfcrossed, i.e., the relation ≺ on the sects of ℑ is irreflexive.
A triangle of ℑ is a set of three mutually adjacent sites (Fig. 3a). Based on this definition, two surfels are meshed iff they are in a triangle and make an orientable 2surfel surface. A triangle {u, v, w}, noted by uvw, has 3 edges, noted uv (or vu), vw (or wv), and wu (or uw), respectively. A surfel e is called crossing an edge uv if e = (u, v) or e = (v, u). A surface is called crossing an edge if a surfel in the surface crosses the edge (Fig. 3b). As shown by Proposition 1, a sect crossing an edge of a triangle always passes through the triangle (Fig. 3c).
Proposition 1
( Triangle Proposition ). If a sect of ℑ crosses an edge of a triangle, then it crosses either of the other two edges, but not both.