Directed path graphs

The concept of a line digraph is generalized to that of a directed path graph. The directed path graph $\overrightarrow P_k(D)$ of a digraph D is obtained by representing the directed paths on k vertices of D by vertices. Two vertices are joined by an arc whenever the corresponding directed paths in D form a directed path on k + 1 vertices or form a directed cycle on k vertices in D. Several properties of $\overrightarrow P_k(D)$ are studied, in particular with respect to isomorphism and traversability

On the total variation regularized estimator over a class of tree graphs

We generalize to tree graphs obtained by connecting path graphs an oracle result obtained for the Fused Lasso over the path graph. Moreover we show that it is possible to substitute in the oracle inequality the minimum of the distances between jumps by their harmonic mean. In doing so we prove a lower bound on the compatibility constant for the total variation penalty. Our analysis leverages insights obtained for the path graph with one branch to understand the case of more general tree graphs. As a side result, we get insights into the irrepresentable condition for such tree graphs.Comment: 42 page

Special asteroidal quadruple on directed path graph non rooted path graph

An asteroidal triple in a graph G is a set of three non- adjacent vertices such that for any two of them there exists a path be- tween them that does not intersect the neighborhood of the third. Special asteroidal triple in a graph G is an asteroidal triple such that each pair is linked by a special connection. A special asteroidal triples play a central role in a characterization of directed path graphs by Cameron, Hoáng and Lévêque. They also introduce a related notion of asteroidal quadru- ple and conjecture a characterization of rooted path graphs. In its original form this conjecture is not complete, still in leafage four, as was showed in. But, as suggested by the conjecture, a characterization by forbidding particular types of asteroidal quadruples may hold. We prove that the conjecture in the original form is true on directed path graphs with leafage four having two minimal separators with multiplicity two. Thus we build the family of forbidden subgraphs in this case.Sociedad Argentina de Informática e Investigación Operativ

Special asteroidal quadruple on directed path graph non rooted path graph

An asteroidal triple in a graph G is a set of three non- adjacent vertices such that for any two of them there exists a path be- tween them that does not intersect the neighborhood of the third. Special asteroidal triple in a graph G is an asteroidal triple such that each pair is linked by a special connection. A special asteroidal triples play a central role in a characterization of directed path graphs by Cameron, Hoáng and Lévêque. They also introduce a related notion of asteroidal quadru- ple and conjecture a characterization of rooted path graphs. In its original form this conjecture is not complete, still in leafage four, as was showed in. But, as suggested by the conjecture, a characterization by forbidding particular types of asteroidal quadruples may hold. We prove that the conjecture in the original form is true on directed path graphs with leafage four having two minimal separators with multiplicity two. Thus we build the family of forbidden subgraphs in this case.Sociedad Argentina de Informática e Investigación Operativ

Best and worst case permutations for random online domination of the path

We study a randomized algorithm for graph domination, by which, according to a uniformly chosen permutation, vertices are revealed and added to the dominating set if not already dominated. We determine the expected size of the dominating set produced by the algorithm for the path graph $P_n$ and use this to derive the expected size for some related families of graphs. We then provide a much-refined analysis of the worst and best cases of this algorithm on $P_n$ and enumerate the permutations for which the algorithm has the worst-possible performance and best-possible performance. The case of dominating the path graph has connections to previous work of Bouwer and Star, and of Gessel on greedily coloring the path.Comment: 13 pages, 1 figur

Isomorphisms and traversability of directed path graphs

The concept of a line digraph is generalized to that of a directed path graph. The directed path graph \forw P_k(D) of a digraph $D$ is obtained by representing the directed paths on $k$ vertices of $D$ by vertices. Two vertices are joined by an arc whenever the corresponding directed paths in $D$ form a directed path on $k+1$ vertices or form a directed cycle on $k$ vertices in $D$. In this introductory paper several properties of \forw P_3(D) are studied, in particular with respect to isomorphism and traversability. In our main results, we characterize all digraphs $D$ with \forw P_3(D)\cong D, we show that \forw P_3(D_1)\cong\forw P_3(D_2) ``almost always'' implies $D_1\cong D_2$, and we characterize all digraphs with Eulerian or Hamiltonian \forw P_3-graphs