Intersection graphs of paths in a tree
Journal of Combinatorial Theory Series B
Theory of linear and integer programming
Theory of linear and integer programming
An almost linear-time algorithm for graph realization
Mathematics of Operations Research
The arborescence-realization problem
Discrete Applied Mathematics
Handbook of combinatorics (vol. 1)
Graph classes: a survey
Recognizing clique graphs of directed and rooted path graphs
Proceedings of the third international conference on Graphs and optimization
Combinatorial optimization: packing and covering
Combinatorial optimization: packing and covering
Recognizing clique graphs of directed edge path graphs
Discrete Applied Mathematics
Graphs and Hypergraphs
Mathematical models to reconstruct phylogenetic trees under the minimum evolution criterion
Networks - Special Issue on Trees
Edge contraction and edge removal on iterated clique graphs
Discrete Applied Mathematics
The maximum vertex coverage problem on bipartite graphs
Discrete Applied Mathematics
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We present a unifying procedure for recognizing intersection graphs of Helly families of paths in a tree and their clique graphs. The Helly property makes it possible to look at these recognition problems as variants of the Graph Realization Problem, namely, the problem of recognizing Edge-Path-Tree matrices. Our result heavily relies on the notion of pie introduced in [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, Journal of Combinatorial Theory, Series B 38 (1985) 8-22] and on the observation that Helly Edge-Path-Tree matrices form a self-dual class of Helly matrices. Coupled to the notion of reduction presented in the paper, these facts are also exploited to reprove and slightly refine some known results for Edge-Path-Tree graphs.