On strongly chordal graphs that are not leaf powers

A common task in phylogenetics is to find an evolutionary tree representing proximity relationships between species. This motivates the notion of leaf powers: a graph G = (V, E) is a leaf power if there exist a tree T on leafset V and a threshold k such that uv is an edge if and only if the distance between u and v in T is at most k. Characterizing leaf powers is a challenging open problem, along with determining the complexity of their recognition. This is in part due to the fact that few graphs are known to not be leaf powers, as such graphs are difficult to construct. Recently, Nevries and Rosenke asked if leaf powers could be characterized by strong chordality and a finite set of forbidden subgraphs. In this paper, we provide a negative answer to this question, by exhibiting an infinite family \G of (minimal) strongly chordal graphs that are not leaf powers. During the process, we establish a connection between leaf powers, alternating cycles and quartet compatibility. We also show that deciding if a chordal graph is \G-free is NP-complete, which may provide insight on the complexity of the leaf power recognition problem

Linear-Time Recognition of Double-Threshold Graphs

A graph G=(V, E) is a double-threshold graph if there exist a vertex-weight function w:V→ℝ and two real numbers lb, ub ∈ ℝ such that uv ∈ E if and only if lb ≤ w(u)+w(v) ≤ ub. In the literature, those graphs are studied also as the pairwise compatibility graphs that have stars as their underlying trees. We give a new characterization of double-threshold graphs that relates them to bipartite permutation graphs. Using the new characterization, we present a linear-time algorithm for recognizing double-threshold graphs. Prior to our work, the fastest known algorithm by Xiao and Nagamochi [Algorithmica 2020] ran in O(n³ m) time, where n and m are the numbers of vertices and edges, respectively

Domain discovery method for topological profile searches in protein structures

We describe a method for automated domain discovery for topological profile searches in protein structures. The method is used in a system TOPStructure for fast prediction of CATH classification for protein structures (given as PDB files). It is important for profile searches in multi-domain proteins, for which the profile method by itself tends to perform poorly. We also present an O(C(n)k +nk2) time algorithm for this problem, compared to the O(C(n)k +(nk)2) time used by a trivial algorithm (where n is the length of the structure, k is the number of profiles and C(n) is the time needed to check for a presence of a given motif in a structure of length n). This method has been developed and is currently used for TOPS representations of protein structures and prediction of CATH classification, but may be applied to other graph-based representations of protein or RNA structures and/or other prediction problems. A protein structure prediction system incorporating the domain discovery method is available at http://bioinf.mii.lu.lv/tops/

Fault-tolerant round robin A/D converter system

Includes bibliographical references (leaves 54-55).Research supported in part by the Draper Laboratories. DL-H-404158 Research supported in part by a Rockwell Doctoral Fellowship. Research supported in part by the Advanced Research Projects Agency, monitored by the Office of Naval Research. N00014-89-J-1489Paul E. Beckmann and Bruce R. Musicus

Algorithms for outerplanar graph roots and graph roots of pathwidth at most 2

Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial effort has been dedicated to deciding whether a given graph has a square root that belongs to a particular graph class. There are both polynomial-time solvable and NP-complete cases, depending on the graph class. We contribute with new results in this direction. Given an arbitrary input graph G, we give polynomial-time algorithms to decide whether G has an outerplanar square root, and whether G has a square root that is of pathwidth at most 2