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Assign one genotype random variable to each person in a pedigree. Independent founders in a pedigree carry population genotype laws, and each other genotype has its conditional distribution given its two parents from Mendelian segregation. The ancestry Directed acyclic graph is a Bayesian network for the joint genotype law. Its moral graph contains each parent-child edge and each coparent edge. Each founder factor and each child-with-parents factor belongs to a clique, giving an undirected graphical representation. The graph is an independence representation; equality with the minimal conditional independence graph needs additional faithfulness assumptions, particularly because inheritance can produce zero probabilities.

Project scheduling asks how quickly a collection of dependent activities can finish. Adding their durations usually overestimates the answer, because independent activities can run together; taking only the longest single activity usually underestimates it, because some activities must wait. The critical path method identifies the chain of dependencies that determines the finish time. Its simplest model assumes known, fixed durations, unlimited parallel processing, and precedence requirements saying that each predecessor must finish before its successor starts. These are the assumptions of the basic model in Hendrickson, Project Management for Construction, §10.2.

Represent each activity by a vertex of a finite Directed acyclic graph; an edge $i\to j$ means that activity $j$ waits for activity $i$. Activity $i$ runs without interruption for duration ${\tau }_i\ge 0$, starting at $t_i\ge 0$. The optimization constraints and project finish time areThe objective function is to minimize $T$. All durations and start times use the same time unit, and the project starts at time zero. The Directed acyclic graph assumption allows a topological order in which every predecessor is processed first.

Consider this small, abstract project:

Activities A and E may start together. After A finishes, B and C may run together, but D must wait for both. F must similarly wait for both D and E: a merge means all incoming requirements must hold. This activity-as-vertex convention keeps durations on the activities and uses arrows only to express precedence.

Unlimited parallel processing is essential to the equality between the longest graph path duration and the optimal finish time. For example, two independent chains, each containing two activities of duration 3, have longest graph path duration 6. Two processors can finish both chains in 6, but one processor needs 12 because all four activities must run sequentially. Thus with limited resources the longest graph path still gives a lower bound, but it need not be achievable. Such resource requirements belong to the broader subject of scheduling.

// Target: mathematical-optimization.bigb

Two Directed acyclic graphs are Markov equivalent directed acyclic graphs if and only if they have the same skeleton of a directed graph and the same unshielded colliders. Thus these two structures determine the observational equivalence class. This structural theorem is what converts the skeleton and collider phases of the PC algorithm into identification of an equivalence class.

Evaluate numeric formula dependencies using depth-first search and memoization, or a topological ordering. Marking active evaluations detects cycles; repeated numerical sweeps need not terminate or determine a unique result.

A spreadsheet dependency graph turns a page of spreadsheet formulas into a plan: compute each prerequisite before the spreadsheet cell that consumes it. This section uses a deliberately small model with explicit cell references, fixed spreadsheet formulas, and deterministic computation from numerical inputs. There are no external data changes or formulas that change their references while running. Within these assumptions, a topological order gives a reproducible answer without repeatedly sweeping the visible grid.

Take two input spreadsheet cells and five spreadsheet formulas:

Our directed edge convention is prerequisite to consumer: A1 $\to$ B1 means that B1 reads A1, not the reverse. The seven directed edges are A1 $\to$ B1, A2 $\to$ B1, A1 $\to$ B2, B1 $\to$ C1, B2 $\to$ C1, C1 $\to$ D1, and A2 $\to$ E1. This spreadsheet dependency graph is a Directed acyclic graph. One topological order is A1,A2,B1,B2,C1,D1,E1; E1 could instead be computed immediately after A2. A topological order constrains prerequisites, not the physical positions of spreadsheet cells.

In that order, the initial arithmetic operations give $B1=3+4=7$, $B2=2\cdot 3=6$, $C1=7+6=13$, $D1=13/2$, and $E1=4^2=16$. The spreadsheet formulas define one value per spreadsheet cell: an outline by induction along any topological order shows that every prerequisite already has its uniquely determined value when its consumer is computed. Different permitted orders therefore agree in this model. The useful idea is that information flows through the Directed acyclic graph even when the visible grid does not display that flow.

A concrete implementation history illustrates this separation. Microsoft's documentation distinguishes the per-worksheet calculation chains used in Excel 2002 and earlier from calculation using multiple software threads introduced in Excel 2007 for independent chain sections. This is a documented implementation example, not a definition of spreadsheet dependency evaluation.^[ref]

Task float, also called total float or task slack, measures an activity's available start-time delay relative to its earliest start, while allowing a feasible schedule that meets the fixed project finish. For earliest start $E_i$, latest start $L_i$, and fixed duration ${\tau }_i$, it isIn the six-activity project scheduling example with start zero and finish 10, B has task float 2 and E has task float 6. B can move from the interval $[3,5]$ to $[5,7]$, and E from $[0,3]$ to $[6,9]$, without changing D's or F's earliest starts. A, C, D, and F have zero task float. With the finish fixed to the optimal duration and the basic critical path method assumptions, zero task float identifies exactly the activities lying on at least one critical graph path. See Hendrickson, §10.4, Activity Float and Schedules.

Task float is not generally an independent allowance that can be spent separately on every activity. For a small example, let two duration-1 activities X then Y form one chain, running in parallel with a separate duration-4 activity. The optimal finish is 4. X has earliest/latest starts 0/2 and Y has earliest/latest starts 1/3, so each has task float 2. Delaying X to 2 forces Y to start at 3, using Y's allowance too. Choosing X at its latest start 2 and Y at its earliest start 1 violates precedence, although both chosen times lie inside their individual ranges. The optimization constraints still couple the decisions. Here B and E can both use their full task floats together because they lie on separate branches; that freedom must be checked from the Directed acyclic graph.

// Target: mathematical-optimization.bigb

A topological order of a finite directed graph $G=(V,E)$ is a sequence $(v_1,\dots ,v_n)$ containing every vertex exactly once, withThus every directed edge points forward in the list. Equivalently, a bijection $p:V\to \{1,\dots ,n\}$ assigning positions satisfies $p(u)<p(v)$ for every directed edge $u\to v$. A topological ordering names the same kind of list. ^[ref]

The toy Directed acyclic graph has exactly five topological orders: $ABCDE$, $ABDCE$, $ACBDE$, $BACDE$, and $BADCE$. These lists were checked by enumerating all $5!=120$ permutations and retaining precisely those satisfying all five directed edge inequalities. Every list ends with $E$, since all other tasks have a directed path to publication. A topological order need not put adjacent entries next to one another in the directed graph: $A$ and $B$ are consecutive in $ABCDE$ without a directed edge between them.

A directed cycle would force the impossible chain of position inequalities $p(u_1)<p(u_2)<\cdots <p(u_k)<p(u_1)$. In the toy example, adding $E\to C$ requires review before publication and publication before review, so there are no topological orders. Together with the construction in topological sorting, this proves that a finite directed graph admits a topological order if and only if it is a Directed acyclic graph. The empty directed graph has the empty sequence as its topological order.

There is also a useful connection to partial orders: define $u⪯v$ when $u=v$ or there is a directed path from $u$ to $v$. For a Directed acyclic graph, this reachability relation is a partial order, and its linear extensions are exactly the topological orders. The Directed acyclic graph specifies which pairs must be ordered; a topological order chooses positions for all the remaining pairs.

A topological ordering is a topological order: a list containing every vertex once and placing the source of every directed edge before its destination. In the parent-and-child language of a Directed acyclic graph, every parent precedes every child, and hence every ancestor precedes every descendant along a directed path. This is an ordering of the vertices, whereas topological sorting is the task of finding it. See topological order for the precise position condition and Kahn's algorithm for a construction.

A topological sorting is the task of constructing a topological order of a finite directed graph, or reporting that a directed cycle makes one impossible. Kahn's algorithm repeatedly takes a vertex with no unfinished prerequisites. This is useful whenever an operation consumes results produced by earlier operations: once a vertex is reached in a topological order, every prerequisite along a directed path has already been processed. ^[ref]

The reason a choice is always available in a nonempty finite Directed acyclic graph is simple. If every vertex had an incoming directed edge, repeatedly following predecessors would eventually revisit a vertex, giving a directed cycle. Removing a vertex and its incident directed edges cannot create a directed cycle, so the argument repeats until every vertex is listed. This proves existence of a topological order and explains why finiteness matters to this procedure.

// Target: combinatorics.bigb

An unshielded collider in a Directed acyclic graph is a triple $a\to b\leftarrow c$ with $a$ and $c$ nonadjacent. The absence of an edge between its outer vertices makes this orientation detectable from conditional independence information under faithfulness of a directed acyclic graph.