In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle, or (for nontrivial cacti) in which every block (maximal subgraph without a cut-vertex) is an edge or a cycle.

Cacti are outerplanar graphs. Every pseudotree is a cactus. A nontrivial graph is a cactus if and only if every block is either a simple cycle or a single edge.

The family of graphs in which each component is a cactus is downwardly closed under graph minor operations. This graph family may be characterized by a single forbidden minor, the four-vertex diamond graph formed by removing an edge from the complete graph K₄.

A triangular cactus is a special type of cactus graph such that each cycle has length three and each edge belongs to a cycle. For instance, the friendship graphs, graphs formed from a collection of triangles joined together at a single shared vertex, are triangular cacti. As well as being cactus graphs the triangular cacti are also block graphs and locally linear graphs.

Triangular cactuses have the property that they remain connected if any matching is removed from them; for a given number of vertices, they have the fewest possible edges with this property. Every tree with an odd number of vertices may be augmented to a triangular cactus by adding edges to it, giving a minimal augmentation with the property of remaining connected after the removal of a matching.

The largest triangular cactus in any graph may be found in polynomial time using an algorithm for the matroid parity problem. Since triangular cactus graphs are planar graphs, the largest triangular cactus can be used as an approximation to the largest planar subgraph, an important subproblem in planarization. As an approximation algorithm, this method has approximation ratio 4/9, the best known for the maximum planar subgraph problem.

The algorithm for finding the largest triangular cactus is associated with a theorem of Lovász and Plummer that characterizes the number of triangles in this largest cactus. Lovász and Plummer consider pairs of partitions of the vertices and edges of the given graph into subsets, with the property that every triangle of the graph either has two vertices in a single class of the vertex partition or all three edges in a single class of the edge partition; they call a pair of partitions with this property valid. Then the number of triangles in the largest triangular cactus equals the maximum, over pairs of valid partitions ${\displaystyle {\mathcal {P}}=\{V_{1},V_{2},\dots ,V_{k}\}}$ and ${\displaystyle {\mathcal {Q}}=\{E_{1},E_{2},\dots ,E_{m}\}}$, of

where ${\displaystyle n}$ is the number of vertices in the given graph and ${\displaystyle u_{i}}$ is the number of vertex classes met by edge class ${\displaystyle E_{i}}$.