Towards a splitter theorem for internally 4-connected binary matroids VIII: small matroids

Our splitter theorem for internally 4-connected binary matroids studies pairs of the form (M,N), where N and M are internally 4-connected binary matroids, M has a proper N-minor, and if M' is an internally 4-connected matroid such that M has a proper M'-minor and M' has an N-minor, then |E(M)|-|E(M')|>3. The analysis in the splitter theorem requires the constraint that |E(M)|>15. In this article, we complement that analysis by using an exhaustive computer search to find all such pairs satisfying |E(M)|<16.Comment: Correcting minor error

Size of the Largest Induced Forest in Subcubic Graphs of Girth at least Four and Five

In this paper, we address the maximum number of vertices of induced forests in subcubic graphs with girth at least four or five. We provide a unified approach to prove that every 2-connected subcubic graph on $n$ vertices and $m$ edges with girth at least four or five, respectively, has an induced forest on at least $n-\frac{2}{9}m$ or $n-\frac{1}{5}m$ vertices, respectively, except for finitely many exceptional graphs. Our results improve a result of Liu and Zhao and are tight in the sense that the bounds are attained by infinitely many 2-connected graphs. Equivalently, we prove that such graphs admit feedback vertex sets with size at most $\frac{2}{9}m$ or $\frac{1}{5}m$, respectively. Those exceptional graphs will be explicitly constructed, and our result can be easily modified to drop the 2-connectivity requirement

Results in lattices, ortholattices, and graphs

This dissertation contains two parts: lattice theory and graph theory. In the lattice theory part, we have two main subjects. First, the class of all distributive lattices is one of the most familiar classes of lattices. We introduce π-versions of five familiar equivalent conditions for distributivity by applying the various conditions to 3-element antichains only. We prove that they are inequivalent concepts, and characterize them via exclusion systems. A lattice L satisfies D0π, if a ✶ (b ✶ c) ≤ (a ✶ b) ✶ c for all 3-element antichains { a, b, c}. We consider a congruence relation ∼ whose blocks are the maximal autonomous chains and define the order- skeleton of a lattice L to be L˜ := L/∼. We prove that the following are equivalent for a lattice L: (i) L satisfies D0π, ( ii) L˜ satisfies any of the five π-versions of distributivity, (iii) the order-skeleton L˜ is distributive. Second, the symmetric difference notion for Boolean algebra is well-known. Matoušek introduced the orthocomplemented difference lattices (ODLs), which are ortholattices associated with a symmetric difference. He proved that the class of ODLs forms a variety. We focus on the class of all ODLs that are set-representable and prove that this class is not locally finite by constructing an infinite set-representable ODL that is generated by three elements. In the graph theory part, we prove generating theorems and splitter theorems for 5-regular graphs. A generating theorem for a certain class of graphs tells us how to generate all graphs in this class from a few graphs by using some graph operations. A splitter theorem tells us how to build up any graph G from any graph H if G contains H. In this dissertation, we find generating theorems for 5-regular graphs and 5-regular loopless graphs for various edge-connectivities. We also find splitter theorems for 5-regular graphs for various edge-connectivities

A survey on algorithmic aspects of modular decomposition

The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important preprocessing step to solve a large number of combinatorial optimization problems. Since the first polynomial time algorithm in the early 70's, the algorithmic of the modular decomposition has known an important development. This paper survey the ideas and techniques that arose from this line of research