Graphs With Small Intersection Dimension Patrick Lillis Advisor: Dr. R. Sritharan

Abstract An X-Graph Split Graphs The boxicity of a graph G, denoted as box(G), We prove that the split graph of a is defined as the minimum integer k such that G is an of axis-parallel k- convex graph has boxicity at most 2, dimensional boxes. We examine some known using intersecting chain graphs. A properties of graphs with respect to boxicity, chain graph is always an , so a 2 chain graph as well as show boxicity results pertaining to Add edge to get B several classes of graphs, including split representation is equivalent to a 2- graphs, X-graphs, and powers of trees. We dimensional box representation. also propose efficient algorithms to produce We then prove than any X-Graph is the intersection of 2 convex graphs, A the relevant k-dimensional representations. Add edge to get A and B (see left). As any convex graph Introduction has boxicity at most 2, any X-graph The graph classes we study all have low then has boxicity at most 4. bounds on boxicity (e.g. a tree has boxicity at most 2), or some result pertaining to small Powers of Trees (left) A tree T, with Δ ≤ 3 boxicity (e.g. it is NP-complete to determine if We find a constant bound on the boxicity (below) An embedding of a split graph has boxicity at most 3). We of powers of trees with Δ at most 3; any T in a revised perfect study specific subclasses of these graph even power of such a tree has binary tree T’. classes in pursuit of generalizable results. boxicity at most 4, any odd We examine those split graphs formed from power has boxicity at most 5. various classes of bipartite graphs by turning This result is shown be first embedding one part into a . Specifically, we find the tree T in a revised perfect binary tree results for split graphs of convex graphs and T’ (see right), and then using the X-graphs. symmetrical structure of such trees to A recent result gave a bound of k+1 on the construct pairs of interval graphs whose boxicity of the kth power of a tree. We intersection is T’. This gives us a result on improve this bound for trees with maximum the boxicity of powers of T’, which in turn (Δ) bounded at 3. can be applied to the original tree T.

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