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Competition Numbers, Quasi-Line Graphs and Holes ; ; Schweitzer, Patrick in SIAM Journal on Discrete Mathematics (in press) The competition graph of an acyclic directed graph D is the undirected graph on the same vertex set as D in which two distinct vertices are adjacent if they have a common out-neighbor in D. The ... [more ▼] The competition graph of an acyclic directed graph D is the undirected graph on the same vertex set as D in which two distinct vertices are adjacent if they have a common out-neighbor in D. The competition number of an undirected graph G is the least number of isolated vertices that have to be added to G to make it the competition graph of an acyclic directed graph. We resolve two conjectures concerning competition graphs. First we prove a conjecture of Opsut by showing that the competition number of every quasi-line graph is at most 2. Recall that a quasi-line graph, also called a locally co-bipartite graph, is a graph for which the neighborhood of every vertex can be partitioned into at most two cliques. To prove this conjecture we devise an alternative characterization of quasi-line graphs to the one by Chudnovsky and Seymour. Second, we prove a conjecture of Kim by showing that the competition number of any graph is at most one greater than the number of holes in the graph. Our methods also allow us to prove a strengthened form of this conjecture recently proposed by Kim, Lee, Park and Sano, showing that the competition number of any graph is at most one greater than the dimension of the subspace of the cycle space spanned by the holes. [less ▲] Detailed reference viewed: 187 (7 UL)Connecting face hitting sets in planar graphs ; Schweitzer, Patrick in Information Processing Letters (2010), 111(1), 11-15 We show that any face hitting set of size n of a connected planar graph with a minimum degree of at least 3 is contained in a connected subgraph of size 5n−6. Furthermore we show that this bound is tight ... [more ▼] We show that any face hitting set of size n of a connected planar graph with a minimum degree of at least 3 is contained in a connected subgraph of size 5n−6. Furthermore we show that this bound is tight by providing a lower bound in the form of a family of graphs. This improves the previously known upper and lower bound of 11n−18 and 3n respectively by Grigoriev and Sitters. Our proof is valid for simple graphs with loops and generalizes to graphs embedded in surfaces of arbitrary genus. [less ▲] Detailed reference viewed: 104 (2 UL) |
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