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A chordal graph is a finite simple graph with no induced cycle of length at least four. Equivalently, every cycle in a graph with at least four vertices has a chord: an edge joining two nonconsecutive vertices of that graph cycle. The word "induced" matters: the induced subgraph on the chosen vertices must contain exactly the cycle's edges, with no chords. Triangles are allowed. This definition concerns adjacency, independently of how the graph is drawn.

For the square cycle graph $C_4$, take vertex set $\{a,b,c,d\}$ and set of edges $\{ab,bc,cd,da\}$, where $ab$ abbreviates $\{a,b\}$. Its graph neighbourhoods are $N(a)=N(c)=\{b,d\}$ and $N(b)=N(d)=\{a,c\}$. The square is an induced cycle, so $C_4$ is not a chordal graph. Add the single edge $ac$: now $N(a)=\{b,c,d\}$, $N(b)=\{a,c\}$, $N(c)=\{a,b,d\}$, and $N(d)=\{a,c\}$. The only possible four-vertex graph cycle has chord $ac$, so this new graph is a chordal graph. Its maximal cliques are exactly $\{a,b,c\}$ and $\{a,c,d\}$.

Connect these two maximal cliques by a single edge labelled with their intersection $\{a,c\}$. The result is a junction tree, also called a clique tree: its nodes are maximal cliques, and the nodes containing any fixed original vertex induce a connected graph inside the tree. Here $a,c$ occur in both nodes, while $b,d$ each occur in just one. This is the running intersection property. More generally, a nonempty connected graph is a chordal graph exactly when its maximal cliques admit such a tree; a disconnected chordal graph can be represented by one such tree per connected component of a graph.^[ref]

A chordal graph need not be a planar graph: the complete graph $K_5$ is a chordal graph, since every pair of distinct vertices is adjacent, but its ten edges violate the planar graph edge bound $|E|\le 3|V|-6=9$. Conversely, even a plane drawing whose bounded faces are triangles can hide an induced cycle: a wheel graph with a four-vertex rim and a central vertex joined to the whole rim still has the rim as an induced cycle. Adding chords to obtain a chordal graph is called triangulation of an undirected graph; this is an adjacency condition, rather than a requirement about planar faces.

Why is this structure useful? The maximal cliques collect local interactions, and the running intersection property makes their overlaps sufficient for consistent communication. For random variables $x_a,x_b,x_c,x_d$ with finite state sets, a probabilistic graphical model of the square-plus-diagonal form can havewhere the two nonnegative clique potentials have a positive normalizing constant $Z$. Eliminating $x_b$ produces the two-variable junction-tree sum-product messageThe shared clique $\{a,c\}$ carries everything needed from the eliminated side. For $q$ possible states per random variable, this example uses two sums over $q^3$ configurations after reusing the message, rather than directly summing over $q^4$ configurations. The example's message is an exact finite sum, with no simulated data. Large maximal cliques still make exact inference expensive; chordal graphs provide a decomposition, not a universal guarantee of cheap inference.^[ref]

A clique in a simple graph $G$ is a set $C$ of vertices such that every pair of distinct members is adjacent. Equivalently, its induced subgraph $G[C]$ is a complete graph. The surrounding graph need not itself be complete: a clique describes complete pairwise connectivity within a selected group.

Consider the original graph withThe set $\{a,b,c\}$ is a clique because all three pairs $ab,ac,bc$ are edges. The set $\{b,c,d\}$ is not a clique: the edge $bd$ is missing. A path in a graph connecting all selected vertices is therefore insufficient; every pair must have a direct edge.

Here the clique number is $3$. The maximum clique is $\{a,b,c\}$, but $\{c,d\}$ and $\{d,e\}$ are also maximal cliques. This distinction is useful when an algorithm stops as soon as its current clique cannot be enlarged: that stopping condition establishes maximality, while finding the largest possible clique is a separate requirement.

A junction tree connects the maximal cliques of a chordal graph into a tree satisfying the running intersection property. Its edges carry separators equal to the intersections of the endpoint cliques. Assigning original factors to containing cliques permits exact sum-product belief propagation without summing independently over the entire collection of variables.

The junction tree algorithm performs exact inference in a finite probabilistic graphical model. First moralize a Bayesian network if necessary; then apply triangulation of an undirected graph, organize its maximal cliques into a junction tree, and assign each original factor to one containing clique. Inward and outward junction-tree sum-product messages compute the normalizing constant and marginal distributions. Its cost depends exponentially on the largest clique's number of variables, rather than necessarily on the total number of variables.

A maximal clique is a clique not strictly contained in another clique. This is distinct from a maximum clique, whose size equals the clique number. In a triangulated probabilistic graphical model, maximal cliques are the natural nodes of a junction tree.

Maximality is a condition on set inclusion: no outside vertex is adjacent to every member of the current clique. In the five-vertex example, $\{c,d\}$ is maximal because $a,b$ are not adjacent to $d$, while $e$ is not adjacent to $c$. Yet $\{a,b,c\}$ is larger. Checking every possible one-vertex extension is enough: if a larger containing clique existed, any one of its new vertices could already be added.^[ref]

A maximum clique is a clique with the greatest possible number of vertices in the graph. Its size is the clique number. Every maximum clique is a maximal clique, since adding a vertex would produce a larger clique. A graph can have several maximum cliques; the word maximum specifies their common size, rather than uniqueness.

The nine maximal cliques of this triangulation areEvery other nonempty clique is a subset of one of these. One junction tree, listing its separators after the edges, isEach variable appears in a connected subtree, verifying the running intersection property. For example, the cliques containing $M$ are $K_1,K_2,K_4,K_5,K_6$, and their connecting paths contain $M$ throughout.

The junction tree algorithm assigns every founder factor and every child-with-parents factor to one containing clique, and multiplies any observation or penetrance factors there. The junction-tree sum-product message from clique $i$ to clique $j$ sums out $C_i\setminus (C_i\cap C_j)$ after multiplying the local potential and all incoming messages except $j$'s. An inward pass sums the joint probability or pedigree likelihood; a subsequent outward pass gives each person's posterior genotype probabilities. This is the same variable-elimination principle as the Elston-Stewart algorithm, organized to handle the pedigree loop. Its largest clique has four genotype variables, much smaller than summing naively over fourteen at once.

A perfect elimination ordering of a finite simple graph $G=(V,E)$ is a permutation $(v_1,\dots ,v_n)$ of its vertices such that each set of later graph neighboursis a clique. Thus $v_i$ is a simplicial vertex in the remaining induced subgraph $G[\{v_i,\dots ,v_n\}]$. Empty and one-vertex sets satisfy the clique condition automatically. The order here is the order of deletion: $v_1$ goes first.

The characterization is powerful: a finite simple graph is a chordal graph if and only if it has a perfect elimination ordering. For one proof outline, the earliest vertex of an induced cycle of length at least four would have two later graph neighbours on that graph cycle that are not adjacent, contradicting the clique condition. Conversely, every nonempty chordal graph has a simplicial vertex, and every induced subgraph of a chordal graph is a chordal graph; repeatedly deleting such a vertex supplies the ordering.^[ref]

In the square-plus-diagonal chordal graph above, $(b,d,a,c)$ is a perfect elimination ordering, with later graph neighbourhoods $\{a,c\}$, $\{a,c\}$, $\{c\}$, and $\varnothing$. Starting instead with $a$ fails: its graph neighbours $b,c,d$ do not form a clique, because $bd$ is absent. Thus even a chordal graph can have a poor elimination ordering.

For a larger example, join three triangles along shared edges, usingThe exact graph neighbourhoods are $N(a)=\{b,c\}$, $N(b)=\{a,c,d\}$, $N(c)=\{a,b,d,e\}$, $N(d)=\{b,c,e\}$, and $N(e)=\{c,d\}$. The ordering $(a,b,c,d,e)$ has later graph neighbourhoods $\{b,c\}$, $\{c,d\}$, $\{d,e\}$, $\{e\}$, and $\varnothing$, all cliques; this certifies that the graph is a chordal graph without enumerating its graph cycles. Its maximal cliques $\{a,b,c\}$, $\{b,c,d\}$, $\{c,d,e\}$ form a junction tree in that order, with intersections $\{b,c\}$ and $\{c,d\}$.

To find an ordering, maximum cardinality search starts with all scores zero, repeatedly selects an unselected vertex of largest score, and increases each unselected graph neighbour's score by one. Reverse the selection order to obtain a candidate perfect elimination ordering. For the three-triangle graph, one valid selection sequence is $(e,d,c,b,a)$, with scores at selection $0,1,2,2,2$; reversing it gives the displayed deletion order. On a chordal graph every tie choice works. On an arbitrary graph, test the resulting later graph neighbourhoods for the clique condition to recognize chordal graphs. Suitable adjacency lists, score buckets and a linear-time verification method give $O(|V|+|E|)$ recognition.^[ref]

The deletion rule also explains sparse Gaussian elimination. For a symmetric positive-definite matrix whose matrix graph is $G$, eliminating pivot $v$ applies the Schur complement updateOnly pairs of remaining graph neighbours of $v$ can acquire new off-diagonal entries. A perfect elimination ordering already has all those edges, so Cholesky decomposition creates no symbolic fill-in. Positive definiteness ensures nonzero pivots in exact arithmetic. Special numerical cancellation can suppress predicted entries, and floating-point rounding affects their values.^[ref] In the three-triangle graph, deleting $c$ first instead connects every pair in $\{a,b,d,e\}$ and adds exactly $ad,ae,be$. The graph was already a chordal graph; the extra fill-in comes from the ordering.

One further reward is optimal graph coloring. Process a perfect elimination ordering backwards and give each vertex the smallest positive color absent from its already colored graph neighbours. Those graph neighbours form a clique, so this greedy algorithm uses at most the clique number $\omega (G)$ colors; any clique of that size requires at least that many. In the three-triangle graph, coloring $(e,d,c,b,a)$ gives colors $(1,2,3,1,2)$, so both the chromatic number and the clique number are three. Every maximal clique also occurs among $\{v_i\}\cup N_i^{+}$: take its earliest vertex and use maximality. The same ordering therefore supports graph coloring, maximal clique extraction and sparse computation.

Historically, Donald J. Rose, Robert E. Tarjan and George S. Lueker's 1976 paper connected graph elimination to symmetric sparse matrices and gave a search-based $O(|V|+|E|)$ algorithm for finding a perfect elimination ordering when one exists. Its abstract distinguishes a minimal elimination ordering from a minimum one: removing avoidable fill-in does not by itself minimize the total number of added edges.^[ref]

Set inclusion $C\subseteq D$ means that every set element of the set $C$ also belongs to the set $D$. Strict set inclusion additionally requires $C\ne D$. A maximal clique cannot grow through strict set inclusion, while a maximum clique maximizes cardinality.