Isomorph-free generation of 2-connected graphs with applications

Many interesting graph families contain only 2-connected graphs, which have ear decompositions. We develop a technique to generate families of unlabeled 2-connected graphs using ear augmentations and apply this technique to two problems. In the first application, we search for uniquely K_r-saturated graphs and find the list of uniquely K_4-saturated graphs on at most 12 vertices, supporting current conjectures for this problem. In the second application, we verifying the Edge Reconstruction Conjecture for all 2-connected graphs on at most 12 vertices. This technique can be easily extended to more problems concerning 2-connected graphs.Comment: 15 pages, 3 figures, 4 table

Characterizing and Detecting Unrevealed Elements of Network Systems

This dissertation addresses the problem of discovering and characterizing unknown elements in network systems. Klir (1985) provides a general definition of a system as “... a set of some things and a relation among the things (p. 4). A system, where the `things\u27, i.e. nodes, are related through links is a network system (Klir, 1985). The nodes can represent a range of entities such as machines or people (Pearl, 2001; Wasserman & Faust, 1994). Likewise, links can represent abstract relationships such as causal influence or more visible ties such as roads (Pearl, 1988, pp. 50-51; Wasserman & Faust, 1994; Winston, 1994, p. 394). It is not uncommon to have incomplete knowledge of network systems due to either passive circumstances, e.g. limited resources to observe a network, active circumstances, e.g. intentional acts of concealment, or some combination of active and passive influences (McCormick & Owen, 2000, p. 175; National Research Council, 2005, pp. 7, 11). This research provides statistical and graph theoretic approaches for such situations, including those in which nodes are causally related (Geiger & Pearl, 1990, pp. 3, 10; Glymour, Scheines, Spirtes, & Kelly, 1987, pp. 75-86, 178183; Murphy, 1998; Verma & Pearl, 1991, pp. 257, 260, 264-265). A related aspect of this research is accuracy assessment. It is possible an analyst could fail to detect a network element, or be aware of network elements, but incorrectly conclude the associated network system structure (Borgatti, Carley, & Krackhardt, 2006). The possibilities require assessment of the accuracy of the observed and conjectured network systems, and this research provides a means to do so (Cavallo & Klir, 1979, p. 143; Kelly, 1957, p. 968)

Reconstructing trees from small cards

The $\ell$-deck of a graph $G$ is the multiset of all induced subgraphs of $G$ on $\ell$ vertices. In 1976, Giles proved that any tree on $n\geq 6$ vertices can be reconstructed from its $\ell$-deck for $\ell \geq n-2$. Our main theorem states that it is enough to have $\ell\geq (8/9+o(1))n$, making substantial progress towards a conjecture of N\'ydl from 1990. In addition, we can recognise connectedness from the $\ell$-deck if $\ell\geq 9n/10$, and reconstruct the degree sequence from the $\ell$-deck if $\ell\ge \sqrt{2n\log(2n)}$. All of these results are significant improvements on previous bounds.Comment: 24 pages, fixed several typo