The intersection of longest paths in a graph.

In this thesis we examine the famous conjecture that every three longest paths in a graph intersect, and add to the classes of graphs for which it is known that this conjecture holds. This conjecture arose from a question asked by Gallai in 1966, the question of whether all of the longest paths in a graph intersect (Gallai's question). In 1969, Walther found a graph in which the longest paths do not all intersect, answering Gallai's question. Since then, many other graphs in which the longest paths do not all intersect have been found. However there are also many classes of graphs for which the longest paths all intersect, such as series-parallel graphs and dually chordal graphs. Finding such classes of graphs is an active area of research and in this thesis we add to these classes of graphs. We begin by investigating Gallai's question for a speci c class of graphs. A theta graph is a graph consisting of three paths with a pair of common endpoints and no other common vertices. A generalised theta graph is a graph with at least one block that consists of at least three paths with a pair of common endpoints and no other common vertices. We show that for a subclass of generalised theta graphs, all of the longest paths intersect. Next, we consider the conjecture that every three longest paths of a graph intersect. We prove that, for every graph with n vertices and at most n + 5 edges, every three longest paths intersect. Finally, we use computational methods to investigate whether all longest paths intersect, or every three longest paths intersect, for several classes of graphs. Two graphs are homeomorphic if each can be obtained from the same graph H by a series of subdivisions. We show that, for every simple connected graph G that is homeomorphic to a simple connected graph with at most 7 vertices, all of the longest paths of G intersect. Additionally, we show that, for every simple connected graph G homeomorphic to a simple connected graph with n vertices, n + 6 edges, and minimum vertex degree 3, all of the longest paths of G intersect. We then show that for every graph with n vertices and at most n + 5 edges, every three longest paths intersect, independently verifying this result. We also present results for several additional classes of graphs with conditions on the blocks, maximum degree of the vertices, and other properties of the graph, showing that every three longest paths intersect or every six longest paths intersect for these graphs

Sublinear Longest Path Transversals and Gallai Families

We show that connected graphs admit sublinear longest path transversals. This improves an earlier result of Rautenbach and Sereni and is related to the fifty-year-old question of whether connected graphs admit constant-size longest path transversals. The same technique allows us to show that $2$-connected graphs admit sublinear longest cycle transversals. We also make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property are exactly the linear forests. Finally, we show that if the order of a connected graph $G$ is large relative to its connectivity $\kappa(G)$ and $\alpha(G) \le \kappa(G) + 2$, then each vertex of maximum degree forms a longest path transversal of size $1$

Intersection of Longest Paths in Graph Theory and Predicting Performance in Facial Recognition

A set of subsets is said to have the Helly property if the condition that each pair of subsets has a non-empty intersection implies that the intersection of all subsets has a non-empty intersection. In 1966, Gallai noticed that the set of all longest paths of a connected graph is pairwise intersecting and asked if the set had the Helly property. While it is not true in general, a number of classes of graphs have been shown to have the property. In this dissertation, we show that K4-minor-free graphs, interval graphs, circular arc graphs, and the intersection graphs of spider graphs are classes that have this property. The accuracy of facial recognition algorithms on images taken in controlled conditions has improved significantly over the last two decades. As the focus is turning to more unconstrained or relaxed conditions and toward videos, there is a need to better understand what factors influence performance. If these factors were better understood, it would be easier to predict how well an algorithm will perform when new conditions are introduced. Previous studies have studied the effect of various factors on the verification rate (VR), but less attention has been paid to the false accept rate (FAR). In this dissertation, we study the effect various factors have on the FAR as well as the correlation between marginal FAR and VR. Using these relationships, we propose two models to predict marginal VR and demonstrate that the models predict better than using the previous global VR

Longest Path and Cycle Transversal and Gallai Families

A longest path transversal in a graph G is a set of vertices S of G such that every longest path in G has a vertex in S. The longest path transversal number of a graph G is the size of a smallest longest path transversal in G and is denoted lpt(G). Similarly, a longest cycle transversal is a set of vertices S in a graph G such that every longest cycle in G has a vertex in S. The longest cycle transversal number of a graph G is the size of a smallest longest cycle transversal in G and is denoted lct(G). A Gallai family is a family of graphs whose connected members have longest path transversal number 1. In this paper we find several Gallai families and give upper bounds on lpt(G) and lct(G) for general graphs and chordal graphs in terms of |V(G)|

Nonempty intersection of longest paths in a graph with a small matching number

summary:A maximum matching of a graph $G$ is a matching of $G$ with the largest number of edges. The matching number of a graph $G$, denoted by $\alpha '(G)$, is the number of edges in a maximum matching of $G$. In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Although this conjecture has been disproved, finding some nice classes of graphs that support this conjecture is still very meaningful and interesting. In this short note, we prove that Gallai's conjecture is true for every connected graph $G$ with $\alpha '(G)\leq 3$

INTERSECTION OF THREE DETOUR PATHS IN GRAPH

Diplomsko delo obravnava problem preseka najdaljših poti v grafu. Poseben poudarek je na preseku treh najdaljših poti, kateremu je namenjeno četrto poglavje. V prvem delu so zapisane osnovne definicije s področja teorije grafov, ki se uporabljajo v nadaljevanju. V naslednjem poglavju se najprej dokaže nepraznost preseka dveh najdaljših poti, nato pa se presek iz dveh najdaljših poti posploši na presek n najdaljših poti. Podanih je nekaj grafov s praznim presekom najdaljših poti. V zadnjem delu poglavja se dokaže nepraznost preseka za sledljiv, hiposledljiv in razcepljen graf. Sledi poglavje, v katerem se osredotočimo na presek najdaljših poti v posameznih blokih grafa. Dokaže se, da je presek najdaljših poti v grafu neprazen natanko tedaj, ko je neprazen presek v vseh blokih grafa. Zadnje poglavje je namenjeno preseku treh najdaljših poti. Podan je tudi dokaz o nepraznosti preseka treh najdaljših poti v zunanje ravninskih grafih.The problem wheteher the intersection of longest paths in a graph is always nonempty is studied. A special emphasis is given on the intersection of three longest ways, this is treated in Section 4. The first part contains basic definitions from the area of graph theory that are needed later. In the next chapter it is first proved that the intersection of two longest paths is always nonempty, and then the problem is generalized to the intersection of more longest paths. Examples of graphs with empty intersection of the set of all longest path are given. On the other hand, the nonemptiness of the intersection is proved for traceable, hypotraceable, and split graphs. In Chapter 3 the focus is on the intersection of longest paths in graphs that contain cut vertices. It is proved that the intersection is nonempty exactly when the intersection is nonempty in all blocks of the graph. The final chapter is devoted to intersections of three longest paths. The main results asserts that three longest paths always intersect provided that they induce an outer planar graphs