Square-Root Finding Problem In Graphs, A Complete Dichotomy Theorem
نویسندگان
چکیده
Graph G is the square of graph H if two vertices x, y have an edge in G if and only if x, y are of distance at most two in H. Given H it is easy to compute its square H. Determining if a given graph G is the square of some graph is not easy in general. Motwani and Sudan [11] proved that it is NP-complete to determine if a given graph G is the square of some graph. The graph introduced in their reduction is a graph that contains many triangles and is relatively dense. Farzad et al. [5] proved the NP-completeness for finding a square root for girth 4 while they gave a polynomial time algorithm for computing a square root of girth at least six. Adamaszek and Adamaszek [1] proved that if a graph has a square root of girth six then this square root is unique up to isomorphism. In this paper we consider the characterization and recognition problem of graphs that are square of graphs of girth at least five. We introduce a family of graphs with exponentially many non-isomorphic square roots, and as the main result of this paper we prove that the square root finding problem is NP-complete for square roots of girth five. This proof is providing the complete dichotomy theorem for square root problem in terms of the girth of the square roots. 1 ar X iv :1 21 0. 76 84 v1 [ cs .D M ] 2 9 O ct 2 01 2
منابع مشابه
Squares of $3$-sun-free split graphs
The square of a graph G, denoted by G, is obtained from G by putting an edge between two distinct vertices whenever their distance is two. Then G is called a square root of G. Deciding whether a given graph has a square root is known to be NP-complete, even if the root is required to be a split graph, that is, a graph in which the vertex set can be partitioned into a stable set and a clique. We...
متن کاملA unified approach to recognize squares of split graphs
The square of a graph G, denoted by G, is obtained from G by putting an edge between two distinct vertices whenever their distance is two. Then G is called a square root of G. Deciding whether a given graph has a square root is known to be NP-complete, even if the root is required to be a split graph, that is, a graph in which the vertex set can be partitioned into a stable set and a clique. We...
متن کاملComputing Graph Roots Without Short Cycles
Graph G is the square of graph H if two vertices x, y have an edge in G if and only if x, y are of distance at most two in H . Given H it is easy to compute its square H, however Motwani and Sudan proved that it is NP-complete to determine if a given graph G is the square of some graph H (of girth 3). In this paper we consider the characterization and recognition problems of graphs that are squ...
متن کاملThe complexity of signed graph and edge-coloured graph homomorphisms
The goal of this work is to study homomorphism problems (from a computational point of view) on two superclasses of graphs: 2-edge-coloured graphs and signed graphs. On the one hand, we consider the H-Colouring problem when H is a 2-edge-coloured graph, and we show that a dichotomy theorem would imply the dichotomy conjecture of Feder and Vardi. On the other hand, we prove a dichotomy theorem f...
متن کاملFinding Cut-Vertices in the Square Roots of a Graph
The square of a given graph H = (V,E) is obtained from H by adding an edge between every two vertices at distance two in H. Given a graph class H, the H-Square Root problem asks for the recognition of the squares of graphs in H. In this paper, we answer positively to an open question of [Golovach et al., IWOCA’16] by showing that the squares of cactus block graphs can be recognized in polynomia...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- CoRR
دوره abs/1210.7684 شماره
صفحات -
تاریخ انتشار 2012