A graph with n vertices is well covered if every maximal independent set is a maximum independent set and very well covered if every maximal independent set has size n/2. In this work, we study these graphs from an algorithmic complexity point of view. We show that well-covered graph recognition is co-NP-complete and that several other problems are NP-complete for well-covered graphs. A number of these problems remain NP-complete on very well covered graphs, while some admit polynomial time solutions for the smaller class. For both families, the isomorphism problem is as hard as general graph isomorphism. DEFINITIONS AND NOTATIONA graph is a pair G = ( V , E ) , where V is a finite set of vertices and E is a set of unordered pairs (u,u) of distinct vertices of V. Each such pair is called an edge.In what follows, G will denote a simple, undirected, finite graph of order n = (VI. Two vertices u and u are adjacent if (UJ) E E. The degree d(u) of a vertex u is the number of vertices adjacent to u. The neighborhood T(u) of u is the set of vertices that are adjacent to u. Two edges are adjacent if they have a vertex in common. A vertex of degree one is called a leaf. An edge that is incident with a leaf is called a pendant edge.A set of vertices is independent if no two of them are adjacent. A graph is a clique if every two vertices are adjacent. A set of vertices in G forms a vertex couer for G if every edge in G is incident on at least one vertex in the set. A subset of E is a matching if no two edges of the set are adjacent. A perfect matching is one in which every vertex in G is an end point of some edge in the matching. A set S is a maximal set satisfying a certain property P if there is no other set properly containing S that satisfies property P. Set S is maximum if there exists no set of greater cardinality that satisfies property P. A similar distinction is made between minimal and minimum. The size of a maximum independent set in a graph G is referred to as cr(G). A graph G is a bipartite graph if V can be partitioned into two independent sets X and Y. We will write the bipartite graph as (X, Y,E). For any undefined terms, see [Z].
In recent years, searching the web on mobile devices has become enormously popular. Because mobile devices have relatively small screens and show fewer search results, search behavior with mobile devices may be different from that with desktops or laptops. Therefore, examining these differences may suggest better, more efficient designs for mobile search engines. In this experiment, we use eye tracking to explore user behavior and performance. We analyze web searches with 2 task types on 2 differently sized screens: one for a desktop and the other for a mobile device. In addition, we examine the relationships between search performance and several search behaviors to allow further investigation of the differences engendered by the screens. We found that users have more difficulty extracting information from search results pages on the smaller screens, although they exhibit less eye movement as a result of an infrequent use of the scroll function. However, in terms of search performance, our findings suggest that there is no significant difference between the 2 screens in time spent on search results pages and the accuracy of finding answers. This suggests several possible ideas for the presentation design of search results pages on small devices.
Compared to the early versions of smart phones, recent mobile devices have bigger screens that can present more web search results. Several previous studies have reported differences in user interaction between conventional desktop computer and mobile device‐based web searches, so it is imperative to consider the differences in user behavior for web search engine interface design on mobile devices. However, it is still unknown how the diversification of screen sizes on hand‐held devices affects how users search. In this article, we investigate search performance and behavior on three different small screen sizes: early smart phones, recent smart phones, and phablets. We found no significant difference with respect to the efficiency of carrying out tasks, however participants exhibited different search behaviors: less eye movement within top links on the larger screen, fast reading with some hesitation before choosing a link on the medium, and frequent use of scrolling on the small screen. This result suggests that the presentation of web search results for each screen needs to take into account differences in search behavior. We suggest several ideas for presentation design for each screen size.
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