Define the following:
(i) Adjacency matrix
(ii) Cut-Set
(iii) Planar graphy
Answers
Answer:
1)In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simple graph, the adjacency matrix is a-matrix with zeros on its diagonal
2)In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one endpoint in each subset of the partition. These edges are said to cross the cut.
3)
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other.[1][2] Such a drawing is called a plane graph or planar embedding of the graph
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Answer:
ɪɴ ɢʀᴀᴘʜ ᴛʜᴇᴏʀʏ ᴀɴᴅ ᴄᴏᴍᴘᴜᴛᴇʀ sᴄɪᴇɴᴄᴇ, ᴀɴ ᴀᴅᴊᴀᴄᴇɴᴄʏ ᴍᴀᴛʀɪx ɪs ᴀ sǫᴜᴀʀᴇ ᴍᴀᴛʀɪx ᴜsᴇᴅ ᴛᴏ ʀᴇᴘʀᴇsᴇɴᴛ ᴀ ғɪɴɪᴛᴇ ɢʀᴀᴘʜ. ᴛʜᴇ ᴇʟᴇᴍᴇɴᴛs ᴏғ ᴛʜᴇ ᴍᴀᴛʀɪx ɪɴᴅɪᴄᴀᴛᴇ ᴡʜᴇᴛʜᴇʀ ᴘᴀɪʀs ᴏғ ᴠᴇʀᴛɪᴄᴇs ᴀʀᴇ ᴀᴅᴊᴀᴄᴇɴᴛ ᴏʀ ɴᴏᴛ ɪɴ ᴛʜᴇ ɢʀᴀᴘʜ. ɪɴ ᴛʜᴇ sᴘᴇᴄɪᴀʟ ᴄᴀsᴇ ᴏғ ᴀ ғɪɴɪᴛᴇ sɪᴍᴘʟᴇ ɢʀᴀᴘʜ, ᴛʜᴇ ᴀᴅᴊᴀᴄᴇɴᴄʏ ᴍᴀᴛʀɪx ɪs ᴀ-ᴍᴀᴛʀɪx ᴡɪᴛʜ ᴢᴇʀᴏs ᴏɴ ɪᴛs ᴅɪᴀɢᴏɴᴀʟ.
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ᴅᴇғɪɴɪᴛɪᴏɴ. ᴀ ᴄᴜᴛ ɪs ᴀ ᴘᴀʀᴛɪᴛɪᴏɴ ᴏғ ᴏғ ᴀ ɢʀᴀᴘʜ ɪɴᴛᴏ ᴛᴡᴏ sᴜʙsᴇᴛs s ᴀɴᴅ ᴛ. ᴛʜᴇ ᴄᴜᴛ-sᴇᴛ ᴏғ ᴀ ᴄᴜᴛ ɪs ᴛʜᴇ sᴇᴛ. ᴏғ ᴇᴅɢᴇs ᴛʜᴀᴛ ʜᴀᴠᴇ ᴏɴᴇ ᴇɴᴅᴘᴏɪɴᴛ ɪɴ s ᴀɴᴅ ᴛʜᴇ ᴏᴛʜᴇʀ ᴇɴᴅᴘᴏɪɴᴛ ɪɴ ᴛ. ɪғ s ᴀɴᴅ ᴛ ᴀʀᴇ sᴘᴇᴄɪғɪᴇᴅ ᴠᴇʀᴛɪᴄᴇs ᴏғ ᴛʜᴇ ɢʀᴀᴘʜ ɢ, ᴛʜᴇɴ ᴀɴ s–ᴛ ᴄᴜᴛ ɪs ᴀ ᴄᴜᴛ ɪɴ ᴡʜɪᴄʜ s ʙᴇʟᴏɴɢs ᴛᴏ ᴛʜᴇ sᴇᴛ s ᴀɴᴅ ᴛ ʙᴇʟᴏɴɢs ᴛᴏ ᴛʜᴇ sᴇᴛ ᴛ.
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ɪɴ ɢʀᴀᴘʜ ᴛʜᴇᴏʀʏ, ᴀ ᴘʟᴀɴᴀʀ ɢʀᴀᴘʜ ɪs ᴀ ɢʀᴀᴘʜ ᴛʜᴀᴛ ᴄᴀɴ ʙᴇ ᴇᴍʙᴇᴅᴅᴇᴅ ɪɴ ᴛʜᴇ ᴘʟᴀɴᴇ, ɪ.ᴇ., ɪᴛ ᴄᴀɴ ʙᴇ ᴅʀᴀᴡɴ ᴏɴ ᴛʜᴇ ᴘʟᴀɴᴇ ɪɴ sᴜᴄʜ ᴀ ᴡᴀʏ ᴛʜᴀᴛ ɪᴛs ᴇᴅɢᴇs ɪɴᴛᴇʀsᴇᴄᴛ ᴏɴʟʏ ᴀᴛ ᴛʜᴇɪʀ ᴇɴᴅᴘᴏɪɴᴛs. ɪɴ ᴏᴛʜᴇʀ ᴡᴏʀᴅs, ɪᴛ ᴄᴀɴ ʙᴇ ᴅʀᴀᴡɴ ɪɴ sᴜᴄʜ ᴀ ᴡᴀʏ ᴛʜᴀᴛ ɴᴏ ᴇᴅɢᴇs ᴄʀᴏss ᴇᴀᴄʜ ᴏᴛʜᴇʀ.[][] sᴜᴄʜ ᴀ ᴅʀᴀᴡɪɴɢ ɪs ᴄᴀʟʟᴇᴅ ᴀ ᴘʟᴀɴᴇ ɢʀᴀᴘʜ ᴏʀ ᴘʟᴀɴᴀʀ ᴇᴍʙᴇᴅᴅɪɴɢ ᴏғ ᴛʜᴇ ɢʀᴀᴘʜ. ᴀ ᴘʟᴀɴᴇ ɢʀᴀᴘʜ ᴄᴀɴ ʙᴇ ᴅᴇғɪɴᴇᴅ ᴀs ᴀ ᴘʟᴀɴᴀʀ ɢʀᴀᴘʜ ᴡɪᴛʜ ᴀ ᴍᴀᴘᴘɪɴɢ ғʀᴏᴍ ᴇᴠᴇʀʏ ɴᴏᴅᴇ ᴛᴏ ᴀ ᴘᴏɪɴᴛ ᴏɴ ᴀ ᴘʟᴀɴᴇ, ᴀɴᴅ ғʀᴏᴍ ᴇᴠᴇʀʏ ᴇᴅɢᴇ ᴛᴏ ᴀ ᴘʟᴀɴᴇ ᴄᴜʀᴠᴇ ᴏɴ ᴛʜᴀᴛ ᴘʟᴀɴᴇ, sᴜᴄʜ ᴛʜᴀᴛ ᴛʜᴇ ᴇxᴛʀᴇᴍᴇ ᴘᴏɪɴᴛs ᴏғ ᴇᴀᴄʜ ᴄᴜʀᴠᴇ ᴀʀᴇ ᴛʜᴇ ᴘᴏɪɴᴛs ᴍᴀᴘᴘᴇᴅ ғʀᴏᴍ ɪᴛs ᴇɴᴅ ɴᴏᴅᴇs, ᴀɴᴅ ᴀʟʟ ᴄᴜʀᴠᴇs ᴀʀᴇ ᴅɪsᴊᴏɪɴᴛ ᴇxᴄᴇᴘᴛ ᴏɴ ᴛʜᴇɪʀ ᴇxᴛʀᴇᴍᴇ ᴘᴏɪɴᴛs.
ᴇᴠᴇʀʏ ɢʀᴀᴘʜ ᴛʜᴀᴛ ᴄᴀɴ ʙᴇ ᴅʀᴀᴡɴ ᴏɴ ᴀ ᴘʟᴀɴᴇ ᴄᴀɴ ʙᴇ ᴅʀᴀᴡɴ ᴏɴ ᴛʜᴇ sᴘʜᴇʀᴇ ᴀs ᴡᴇʟʟ, ᴀɴᴅ ᴠɪᴄᴇ ᴠᴇʀsᴀ, ʙʏ ᴍᴇᴀɴs ᴏғ sᴛᴇʀᴇᴏɢʀᴀᴘʜɪᴄ ᴘʀᴏᴊᴇᴄᴛɪᴏɴ.
ᴘʟᴀɴᴇ ɢʀᴀᴘʜs ᴄᴀɴ ʙᴇ ᴇɴᴄᴏᴅᴇᴅ ʙʏ ᴄᴏᴍʙɪɴᴀᴛᴏʀɪᴀʟ ᴍᴀᴘs.
ᴛʜᴇ ᴇǫᴜɪᴠᴀʟᴇɴᴄᴇ ᴄʟᴀss ᴏғ ᴛᴏᴘᴏʟᴏɢɪᴄᴀʟʟʏ ᴇǫᴜɪᴠᴀʟᴇɴᴛ ᴅʀᴀᴡɪɴɢs ᴏɴ ᴛʜᴇ sᴘʜᴇʀᴇ ɪs ᴄᴀʟʟᴇᴅ ᴀ ᴘʟᴀɴᴀʀ ᴍᴀᴘ. ᴀʟᴛʜᴏᴜɢʜ ᴀ ᴘʟᴀɴᴇ ɢʀᴀᴘʜ ʜᴀs ᴀɴ ᴇxᴛᴇʀɴᴀʟ ᴏʀ ᴜɴʙᴏᴜɴᴅᴇᴅ ғᴀᴄᴇ, ɴᴏɴᴇ ᴏғ ᴛʜᴇ ғᴀᴄᴇs ᴏғ ᴀ ᴘʟᴀɴᴀʀ ᴍᴀᴘ ʜᴀᴠᴇ ᴀ ᴘᴀʀᴛɪᴄᴜʟᴀʀ sᴛᴀᴛᴜs.
ᴘʟᴀɴᴀʀ ɢʀᴀᴘʜs ɢᴇɴᴇʀᴀʟɪᴢᴇ ᴛᴏ ɢʀᴀᴘʜs ᴅʀᴀᴡᴀʙʟᴇ ᴏɴ ᴀ sᴜʀғᴀᴄᴇ ᴏғ ᴀ ɢɪᴠᴇɴ ɢᴇɴᴜs. ɪɴ ᴛʜɪs ᴛᴇʀᴍɪɴᴏʟᴏɢʏ, ᴘʟᴀɴᴀʀ ɢʀᴀᴘʜs ʜᴀᴠᴇ ɢʀᴀᴘʜ ɢᴇɴᴜs , sɪɴᴄᴇ ᴛʜᴇ ᴘʟᴀɴᴇ (ᴀɴᴅ ᴛʜᴇ sᴘʜᴇʀᴇ) ᴀʀᴇ sᴜʀғᴀᴄᴇs ᴏғ ɢᴇɴᴜs . sᴇᴇ "ɢʀᴀᴘʜ ᴇᴍʙᴇᴅᴅɪɴɢ" ғᴏʀ ᴏᴛʜᴇʀ ʀᴇʟᴀᴛᴇᴅ ᴛᴏᴘɪᴄs.